diff --git "a/F9AzT4oBgHgl3EQfUfyk/content/tmp_files/load_file.txt" "b/F9AzT4oBgHgl3EQfUfyk/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/F9AzT4oBgHgl3EQfUfyk/content/tmp_files/load_file.txt" @@ -0,0 +1,1069 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf,len=1068 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='01268v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='CV] 3 Jan 2023 Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets Barbara Drinovec Drnovˇsek and Franc Forstneriˇc Abstract We show that if E is a closed convex set in Cn (n > 1) contained in a closed halfspace H such that E ∩ bH is nonempty and bounded, then the concave domain Ω = Cn \\ E contains images of proper holomorphic maps f : X → Cn from any Stein manifold X of dimension < n, with approximation of a given map on closed compact subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If in addition 2 dim X + 1 ≤ n then f can be chosen an embedding, and if 2 dim X = n then it can be chosen an immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Under a stronger condition on E we also obtain the interpolation property for such maps on closed complex subvarieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Keywords Stein manifold, holomorphic embedding, Oka manifold, minimal surface, convexity MSC (2010): 32H02, 32Q56;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 52A20, 53A10 Date: 3 January 2023 In memoriam Nessim Sibony 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Introduction Let X be a Stein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Denote by O(X, Cn) the Frechet space of holomorphic maps X → Cn endowed with the compact-open topology and write O(X, C) = O(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' A theorem of Remmert [36] (1956), Narasimhan [35] (1960), and Bishop [7] (1961) states that almost proper maps are residual in O(X, Cn) if dim X = n, proper maps are dense if dim X < n, proper immersions are dense if 2 dim X ≤ n, and proper embeddings are dense if 2 dim X < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' A proof is also given in the monograph [29] by Gunning and Rossi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It is natural to ask how much space proper maps need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We pose the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For which domains Ω ⊂ Cn are proper holomorphic maps (immersions, embeddings) X → Cn as above, with images contained in Ω, dense in O(X, Ω)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It is evident that Ω cannot be contained in a halfspace of Cn since every holomorphic map from C to a halfspace lies in a complex hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In this paper we give an affirmative answer for concave domains whose complement E = Cn \\ Ω satisfies the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' A closed convex set E in a real or complex Euclidean space V has bounded convex exhaustion hulls (BCEH) if for every compact convex set K in V (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) the set h(E, K) = Conv(E ∪ K) \\ E is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Here, Conv denotes the convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The following is our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let E be an unbounded closed convex set in Cn (n > 1) with bounded convex exhaustion hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Given a Stein manifold X with dim X < n, a compact O(X)-convex set K in X, and a holomorphic map f0 : K → Cn with f0(bK) ⊂ Ω = Cn \\ E, we can approximate f0 uniformly on K by proper holomorphic maps f : X → Cn satisfying f(X \\ ˚ K) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The map f can be chosen an embedding if 2 dim X < n and an immersion if 2 dim X ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Forstneriˇc In this paper, a map f : K → Cn from a compact set K is said to be holomorphic if it is the restriction to K of a holomorphic map on an open neighbourhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In particular, if f0(K) ⊂ Ω then the theorem gives uniform approximation of f0 by proper holomorphic maps f : X → Cn with f(X) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If bE is of class C 1 and strictly convex near infinity, we obtain an analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 with additional interpolation on a closed complex subvariety X′ of X such that f0 : X′ → Cn is proper holomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Without the condition on the range, interpolation of proper holomorphic embeddings X ֒→ Cn on a closed complex subvariety was obtained by Acquistapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' [1] in 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The analogue of the BCEH condition for unbounded closed sets in Stein manifolds, with the convex hull replaced by the holomorphically convex hull, is used in holomorphic approximation theory of Arakelyan and Carleman type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see the survey in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It is evident that a closed convex set E ⊂ Rn has BCEH if and only if there is an increasing sequence K1 ⊂ K2 ⊂ · · · of compact convex sets exhausting Rn such that the set h(E, Kj) (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1)) is bounded for every j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='. In particular, BCEH is a condition at infinity which is invariant under perturbations supported on a compact subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For compact convex sets E ⊂ Cn, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 was proved in [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' in this case BCEH trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We show in Section 3 that a closed convex set E in Rn has BCEH if and only if E is continuous in the sense of Gale and Klee [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If E has BCEH then Conv(E ∪ K) is closed for any compact convex set K ⊂ Rn (see [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If such E is unbounded, which is the main case of interest, there are affine coordinates (x, y) ∈ Rn−1 × R such that E = Eφ = {(x, y) ∈ Rn : y ≥ φ(x)} is the epigraph of a convex function φ : Rn−1 → R+ = [0, +∞) growing at least linearly near infinity (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In particular, an unbounded closed convex set E ⊂ Cn with BCEH is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) E = Eφ = {z = (z′, zn) ∈ Cn : ℑzn ≥ φ(z′, ℜzn)} in some affine complex coordinates z = (z′, zn) on Cn, with φ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (Here, ℜ and ℑ denote the real and the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') For a convex function φ of class C 1 we give a differential characterization of the BCEH condition on its epigraph Eφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The BCEH property holds if the radial derivative of φ tends to infinity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' On the other hand, there are convex functions of linear growth whose epigraphs have BCEH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='11, a convex function φ with at least linear growth at infinity can be approximated uniformly on compacts by functions ψ ≤ φ of the same kind whose epigraphs Eψ have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This allows us to extend Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see Section 4 for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The conclusion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 holds for any convex epigraph Eφ of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) such that φ ≥ 0 and the set {φ = 0} is nonempty and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' A closed convex set E ⊂ Cn with BCEH does not contain any affine real line (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4), and for n > 1 its complement Ω = Cn \\ E is an Oka domain according to Wold and the second named author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This fact plays an important role in our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3, given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (The precise result from Oka theory which we shall use is stated as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') Among closed convex epigraphs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2), the class of sets with Oka complement is strictly bigger than the class of sets with BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In particular, the former class contains many sets containing boundary lines, which is impossible for a set with BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Is there a (not necessarily convex) set Eφ ⊂ Cn of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) with φ ≥ 0 of sublinear growth for which Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 holds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Is there a set of this kind in C2 such that C2\\Eφ contains the image of a proper holomorphic disc D = {z ∈ C : |z| < 1} → C2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 is the first general result in the literature providing proper holomorphic maps X → Cn from any Stein manifold of dimension < n whose images avoid large convex sets in Cn close to a halfspace, and with approximation of a given map on a compact holomorphically convex set in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Without the approximation condition and assuming that dim X ≤ n − 2, there are proper holomorphic maps of X into a complex hyperplane in Cn \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' On the other hand, there are many known results concerning proper holomorphic maps in Euclidean spaces and in more general Stein manifolds whose images avoid certain small closed subsets, such as compact or complete pluripolar ones, and results in which the source manifold is the disc D = {z ∈ C : |z| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proper holomorphic discs in C2 avoiding closed complete pluripolar sets of the form E = E′ × C, with E′ ⊂ C, were constructed by Alexander [5] in 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The first named author showed in [13] (2004) that for every closed complete pluripolar set E in a Stein manifold Y with dim Y > 1 and point p ∈ Y \\ E there is a proper holomorphic disc f : D → Y with p ∈ f(D) ⊂ Y \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If Y = C2 there also exist embedded holomorphic discs with this property according to Borell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' [8] (2008), and for dim Y ≥ 3 this holds by the general position argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proper holomorphic discs in C2 with images contained in certain concave cones were constructed by Globevnik and the second named author [23] in 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' They also constructed proper holomorphic discs in C2 with images in (C \\ {0})2, and hence proper harmonic discs D → R2, disproving a conjecture by Schoen and Yau [37, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (Another construction of such maps was given by Boˇzin [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') More generally, it was shown by Alarc´on and L´opez [4, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1] in 2012 that every open Riemann surface X admits a proper harmonic map to R2 which is the projection of a conformal minimal immersion X → R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The aforementioned result from [23] was used by the first named author in [12] (2002) to classify closed convex sets in C2 whose complement is filled by images of holomorphic discs which are proper in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' More recently, Forstneriˇc and Ritter [24] (2014) proved Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 in the case when E ⊂ Cn is a compact polynomially convex set and 2 dim X ≤ n (for immersions) or 2 dim X < n (for embeddings), and for proper holomorphic maps X → Cn when dim X < n and E is a compact convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' A further development in this direction is the analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 when Cn is replaced by a Stein manifold Y with the density property and E ⊂ Y is a compact O(Y )-convex set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see [22, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' However, in all mentioned results except those in [23, 12], the avoided sets are thin or compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Without insisting on approximation, the theorem of Remmert, Bishop, and Narasimhan is not optimal with respect to the dimension of the target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Indeed, it was shown by Eliashberg and Gromov [17] in 1992, with an improvement for odd dimensional Stein manifolds by Sch¨urmann [38] in 1997, that a Stein manifold X of dimension m ≥ 2 embeds properly holomorphically in Cn with n = �3m 2 � + 1, and for m ≥ 1 it immerses properly holomorphically in Cn with n = � 3m+1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (See also [20, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') However, the construction method in these papers, which relies on the Oka principle for sections of certain stratified holomorphic fibre bundles, does not give the density statement, and we do not know whether Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 holds for maps to these lower dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It is also an open problem whether every open Riemann surface embeds properly holomorphically in C2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see [20, Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='10-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='11] and the survey [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 is proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The proof relies on two main ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' One is the result of Wold and the second named author [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='8] which shows in particular that the complement Ω = Cn \\E of a closed convex set E having BCEH is an Oka domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The second main technique comes from the work of Dor [10, 11] (1993-95), following earlier papers by Stensønes [39] (1989) and Hakim [30] (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Dor constructed proper holomorphic immersions and embeddings of any smoothly bounded, relatively compact, strongly pseudoconvex domain D in a Stein manifold X into any pseudoconvex domain Ω in Cn under the dimension conditions 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Forstneriˇc in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Previously, Hakim [30] constructed proper holomorphic maps to balls in codimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The main idea is to inductively lift the image of bD under a holomorphic map f : ¯D → Ω to a given higher superlevel set of a strongly plurisubharmonic exhaustion function ρ : Ω → R+ in a controlled way, taking care not to decrease the value of ρ ◦ f very much anywhere on D during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' When D is a finite bordered Riemann surface, this can be achieved by using approximate solutions of a Riemann-Hilbert boundary value problem (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In higher dimensions the proof is more subtle and uses carefully controlled holomorphic peak functions on ¯D to push a given map f : ¯D → Ω locally at a point z ∈ f(bD) in the direction of the zero set Sz of the holomorphic (quadratic) Levi polynomial of the exhaustion function ρ : Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' At a noncritical point z ∈ Ω of ρ, Sz is a smooth local complex hypersurface and the restricted function ρ|Sz increases quadratically as we move away from z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If ρ is a strictly convex function, this can be achieved by pushing the image of f(bD) in the direction of suitably chosen affine complex hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Dor’s construction was extended by the authors to maps from strongly pseudoconvex domains in Stein manifolds to an arbitrary Stein manifold Ω, and also to q-convex complex manifolds for suitable values of q ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see the papers [14, 15] from 2007 and 2010, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In those papers we introduced the technique of gluing holomorphic sprays of manifold-valued maps on a strongly pseudoconvex Cartan pair with control up to the boundary (a nonlinear version of the Cousin-I problem) and a systematic approach for avoiding critical points of a q-convex Morse exhaustion function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Earlier constructions of this type, using simpler holomorphic peak functions and higher codimension, were given in 1985 by Løw [34] and Forstneriˇc [19] who showed that every relatively compact strongly pseudoconvex domain D in a Stein manifold embeds properly holomorphically in a high dimensional Euclidean ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' A related result with interpolation on a suitable subset of the boundary of D is due to Globevnik [27] (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This peak function technique was inspired by the construction of inner functions on the ball of Cn by Løw [33] in 1982, based on the work of Hakim and Sibony [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We apply this technique to push the boundary f0(bD) ⊂ Ω = Cn \\ E of a holomorphic map f0 : ¯D → Cn in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 out of a certain compact convex cap C attached to E along a part of bC contained in bE and such that the set E1 = E ∪ C is convex and has bounded convex exhaustion hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' At the same time, we ensure that the new map g : ¯D → Cn still sends D \\ K to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For a precise result, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In the next step, we use that Ω1 = Cn \\ E1 is an Oka domain (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since g(bD) ⊂ Ω1, we can apply the Oka principle (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) to approximate g by a holomorphic map f1 : X → Cn with f1(X \\ D) ⊂ Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Continuing inductively, we obtain a sequence of holomorphic maps X → Cn converging to a proper map satisfying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The details are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The analogues of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4 also hold for minimal surfaces in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let n ≥ 3, and let φ : Rn−1 → R+ be a convex function such that the set {φ = 0} is nonempty and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Given an open Riemann surface X, a compact O(X)-convex set K in X, and a conformal minimal immersion f0 : U → Rn from a neighbourhood of K with f0(bK) ⊂ Ω = {y < φ(x)}, we can approximate f0 uniformly on K by proper conformal minimal immersions f : X → Rn (embeddings if n ≥ 5) satisfying f(X \\ ˚ K) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If in addition φ is of class C 1, strictly convex at infinity, and the epigraph Eφ = {y ≥ φ(x)} has BCEH then one can add to this statement the interpolation of the map on discrete sets, in analogy to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6 is obtained by following the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3, replacing Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 by the analogous result obtained by the Riemann–Hilbert deformation method for conformal Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets 5 minimal surfaces (see [2] or [3, Chapter 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Furthermore, it has recently been shown by the authors [16, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5] that the complement of a closed convex set E ⊂ Rn (n ≥ 3) is flexible for minimal surfaces (an analogue of the Oka property in complex geometry) if and only if E is not a halfspace or a slab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' clearly this includes all sets with BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Another method for constructing proper minimal surfaces, which yields the same result in some examples not covered by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6, was developed by Alarc´on and L´opez [4] in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' They showed that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6 holds for any wedge domain Γ × R ⊂ R3, where Γ ⊂ R2 is an open cone with angle > π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see [4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The complement of this set is convex but it fails to satisfy the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6 due to the presence of lines in the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' An important difference between these two fields, which affects the possible construction methods, is that every open Riemann surface admits a proper harmonic map to the plane R2 (see [4, Theorem I]), while only few such surfaces admit proper holomorphic maps to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The analogue of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5 for minimal surfaces asks whether there is a domain in R3 of the form {x3 < φ(x1, x2)}, where φ : R2 → R+ is a function with sublinear growth, which contains minimal surfaces of hyperbolic type that are proper in R3, or just a proper hyperbolic end of a minimal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In particular, it would be interesting to know whether the domain below the upper half of a vertical catenoid has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' On the other hand, the strong halfspace theorem of Hoffman and Meeks [32] says that the only proper minimal surfaces in R3 contained in a halfspace are planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Pushing a strongly pseudoconvex boundary out of a strictly convex cap Let O be a convex domain in Cn for some n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Recall that a continuous function ρ : O → R is said to be strictly convex if for any pair of points a, b ∈ O we have that ρ(ta + (1 − t)b) < tρ(a) + (1 − t)ρ(b) for all 0 < t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume now that ρt : O → R (t ∈ [0, 1]) is a continuous family of C 1 functions satisfying the following conditions: (a) For every t ∈ [0, 1] the function ρt is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that dρt ̸= 0 on Mt := {ρt = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (b) If 0 ≤ s < t ≤ 1 then ρt ≤ 0 on Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (c) There is an open relatively compact subset ω0 of M0 such that for every pair of numbers 0 ≤ s < t ≤ 1 we have that Mt ∩ M0 = Mt ∩ Ms = M0 \\ ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This means that the hypersurfaces Mt coincide on the subset M0 \\ ω0, and as t ∈ [0, 1] increases the domains ωt = Mt \\ M0 ⊂ Mt are pairwise disjoint and move into the convex direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Each compact set of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) Ct = � s∈[0,t] ωs for t ∈ [0, 1] is called a strictly convex cap with the base ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that bCt = ω0 ∪ ωt, Ct is strictly convex along ωt, strictly concave along ω0, and it has corners along ω0 ∩ ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' As t ∈ [0, 1] increases to 1, the caps Ct monotonically increase to C1 and they share the same base ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Likewise, for any 0 ≤ s < t ≤ 1 the set Cs,t = � u∈[s,t] ωu is a strictly convex cap with the base ωs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The sets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) Et = {z ∈ O : ρt(z) ≤ 0} for t ∈ [0, 1] are strictly convex along bEt = {ρt = 0}, they form a continuously increasing family in t, and Et = E0 ∪ Ct for every t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Forstneriˇc Under these assumptions, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let D be a smoothly bounded, relatively compact, strongly pseudoconvex domain in a Stein manifold X with dim X < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let the sets Et ⊂ O ⊂ Cn (t ∈ [0, 1]) be given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2), and let f0 : ¯D → O be a map of class A ( ¯D) such that f0(bD) ∩ E0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Given a compact set K ⊂ D such that f0(D \\ K) ∩ E0 = ∅ and a number ǫ > 0, there is a map f : ¯D → O of class A ( ¯D) satisfying the following conditions: (i) f(bD) ∩ E1 = ∅, (ii) f(D \\ K) ∩ E0 = ∅, and (iii) maxx∈K |f(x) − f0(x)| < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Recall that a map f : ¯D → O is said to be of class A ( ¯D) if it is continuous on ¯D and holomorphic on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In our application of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3, the set O will be a ball (or the entire Euclidean space) and the hypersurfaces Mt = {ρt = 0} = bEt will be convex graphs over the coordinate hyperplane Cn−1 × R ⊂ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 we shall need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume that O is a convex open subset of Cn for n > 1, L is a compact subset of O, and ρ : O → R is a C 1 smooth strictly convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then there is a number δ > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If D is a smoothly bounded strongly pseudoconvex domain in a Stein manifold X of dimension dim X = m < n, K is a compact subset of D, and f : ¯D → O is a map of class A ( ¯D) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) ρ(f(z)) > −δ for all z ∈ bD and ρ(f(z)) > 0 if z ∈ bD and f(z) /∈ L, then given η > 0 there is a map g : ¯D → O of class A ( ¯D) satisfying the following conditions: (i) ρ(g(z)) > 0 for z ∈ bD, (ii) ρ(g(z)) > δ for those z ∈ bD for which g(z) ∈ L, (iii) ρ(g(z)) > ρ(f(z)) − η for z ∈ D \\ K, and (iv) |f(z) − g(z)| < η for z ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For m = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=', when D is a finite bordered Riemann surface, this is a simplified version of [14, Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3], which is proved by using approximate solutions of a Riemann–Hilbert boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This method was employed in several earlier papers mentioned in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' When ρ is strictly convex, C 1 smoothness suffices since in the proof we may take a continuous family of tangential linear discs to the sublevel set of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For m ≥ 2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 is a simplified and slightly modified version of [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Besides the fact that we are considering single maps ¯D → O instead of sprays of maps, the only difference is that the assumption in [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3] that the set {ρ = 0} is compact is replaced by the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) saying that ρ(f(z)) for z ∈ bD may be negative only if f(z) lies in the compact set L ⊂ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This hypothesis ensures that the lifting for a relatively big amount (the role of the constant δ) only needs to be made on a compact subset of O, while elsewhere it suffices to pay attention not to decrease ρ ◦ f by more than a given amount and to approximate sufficiently closely on K (the role of the constant η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The proof requires only a minor adaptation of [15, proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3], using its local version [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2] in a finite induction with respect to a covering of bD by small open sets on which there are good systems of local holomorphic peak functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In fact, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 corresponds to a simplified version of [15, Sublemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4], which explains how to lift the image of bD with respect to ρ for a sufficiently large amount at those points in bD which the map f sends to a certain coordinate chart Ui in the target manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets 7 In our situation, the role of Ui is played by an open relatively compact neighbourhood of the set L ∩ {ρ = 0} in O, and there is no need to use the rest of the proof of [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For t ∈ [0, 1] let δt > 0 be a number for which the conclusion of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 holds for the function ρt and the compact set L = C1 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The open sets Ut = {z ∈ O : −δt < ρt(z) < δt} for t ∈ [0, 1] form an open covering of C1, so there exists a finite subcovering {Utj} for 0 ≤ t1 < t2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' < tk ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 we inductively find maps f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , fk ∈ A ( ¯D) such that for every j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , k we have that (a) fj(bD) ∩ Etj = ∅ (where Et is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2)), (b) fj(D \\ K) ∩ E0 = ∅, and (c) |fj − fj−1| < ǫ/k on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that conditions (a) and (b) hold for f0 and (c) is void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume inductively that for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , k} we have maps f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , fj−1 satisfying these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 with f = fj−1 and taking fj = g, condition (a) follows from part (i) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2, (b) follows from (ii) provided that the number η > 0 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 is chosen small enough, and (c) follows from (iii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 provided that η ≤ ǫ/k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This gives the map fj satisfying conditions (a)–(c) and the induction may continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The map f = fk then satisfies the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 also holds, with the same proof, if ρt (t ∈ [0, 1]) are strongly plurisubharmonic functions of class C 2 satisfying dρt ̸= 0 on Mt = {ρt = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Indeed, the results from [15], which are used in the proof, pertain to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In the present paper we shall only use the convex case under C 1 smoothness, which comes naturally in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Closed convex sets with BCEH In the context of convex analysis, closed unbounded convex sets that share several important properties with compact convex sets were studied by Gale and Klee [26] in 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' They introduced the class of continuous sets, and we show that this class coincides with the class of sets having BCEH, introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We then develop further properties of these sets which are relevant to the proof of our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By a ray in Rn, we shall mean a closed affine halfline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let E be a closed convex subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' A boundary ray of E is a ray contained in the boundary of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' An asymptote of E is a ray L ⊂ Rn \\ E such that dist(L, E) = inf{|x − y| : x ∈ L, y ∈ E} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The function σ : {u ∈ Rn : |u| = 1} → R ∪ {+∞}, σ(u) = sup{x · u : x ∈ E} is called the the support function of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (Here, x · u denotes the Euclidean inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') A closed convex set E is said to be continuous in the sense of Gale and Klee [26] if the support function of E is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that every compact convex set is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The following result is a part of [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3] due to Gale and Klee;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' we only list those conditions that will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The last item (iv) uses also [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For a closed convex subset E in Rn the following conditions are equivalent: (i) E is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (ii) E has no boundary ray nor asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (iii) For each point p ∈ Rn the convex hull Conv(E ∪ {p}) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (iv) For every compact convex set K ⊂ Rn the set Conv(E ∪ K) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Forstneriˇc Condition (iii) implies that the closed convex hull Conv(E ∪ {p}) is the union of the line segments connecting p to the points in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It also shows that an unbounded continuous closed convex subset E of Rn is not contained in any affine hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let us record the following observation which will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let E ⊂ Rn be a closed convex set, p ∈ Rn\\E, and L ⊂ Rn be an affine subspace containing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then, Conv(E ∪ {p}) ∩ L = Conv((E ∩ L) ∪ {p}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Set E′ = E ∩ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It is obvious that Conv(E′ ∪ {p}) ⊂ Conv(E ∪ {p}) ∩ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Conversely, since E is convex, every point q ∈ Conv(E ∪ {p}) belongs to a line segment from p to a point q′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If in addition q ∈ L and q ̸= p then q′ ∈ E′, and hence q ∈ Conv(E′ ∪ {p}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' A closed convex set E ⊂ Rn has BCEH if and only if it is continuous in the sense of Gale and Klee [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since all closed bounded convex sets have BCEH and are continuous, it suffices to consider the case when the set E is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If E is not continuous then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 it has a boundary ray or an asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Denote it by L, and let ℓ be the affine line containing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Pick any affine 2-plane H ⊂ Rn containing ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' There is a point p ∈ H \\(ℓ∪E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By considering rays from p to points q ∈ E approaching L and going to infinity (if L is a boundary ray, we can choose points q ∈ L), we see that the closure of the set h(E, p) = Conv(E ∪ {p}) \\ E contains the parallel translate L′ ⊂ H+ of L passing through p, so h(E, p) is unbounded and hence E does not have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume now that E is a continuous and let us prove that it has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We need to show that for any closed ball B ⊂ Rn the set h(E, B) = Conv(E ∪ B) \\ E is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume to the contrary that there is a sequence xm ∈ h(E, B) with |xm| → ∞ as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since the sets E and B are convex, we have that xm = tmbm + (1 − tm)em for tm ∈ [0, 1], bm ∈ B, em ∈ E, and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that (1 − tm)|em| → ∞ as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By compactness of the respective sets we may assume, passing to a subsequence, that em ̸= 0 for all m and the sequences tm, bm, and 1 |em|em are convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Denote their respective limits by t, b, and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We have that xm = tmbm + (1 − tm)em = bm + (1 − tm)|em| � em |em| − bm |em| � = bm + (1 − tm)|em|fm where fm = � em |em|− bm |em| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that limm→∞ fm = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Pick a number α ≥ 0 and set p = b+αf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If m is large enough then (1−tm)|em| > α, so the point ym = bm+αfm lies on the line segment connecting bm and xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since xm ∈ Conv(E ∪ {bm}), it follows that ym ∈ Conv(E ∪ {bm}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that the sequence ym converges to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since E is continuous, Conv(E ∪ {b}) is closed by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1, so p = limm→∞ ym ∈ Conv(E ∪ {b}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since this holds for every α ≥ 0, the ray L = {b + αf : α ∈ [0, ∞)} lies in Conv(E ∪ {b}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 there is α0 ∈ [0, ∞) such that the ray L′ = {b + αf : α ≥ α0} lies in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since E is continuous, L is not a boundary ray of E by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1, thus L contains a point q = b+α1f ∈ E \\bE for some α1 ≥ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Choose a neighbourhood Uq ⊂ E of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For any large enough m we then have pm := bm + α1fm ∈ Uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let Lm = {bm + αfm : α ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that Lm ∩ Conv(E ∪ {bm}) = Conv((Lm ∩ E) ∪ {bm}) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' However, for m large enough the point xm ∈ Lm lies on the opposite side of pm than bm, so xm belongs to Lm ∩ Conv(E ∪ {bm}) but not to Conv((Lm ∩ E) ∪ {bm}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This contradiction proves that E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets 9 Given a function φ : Rn−1 → R, the epigraph of φ is the set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) E = Eφ = {(x, y) ∈ Rn−1 × R : y ≥ φ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that a function is convex if and only if its epigraph is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If E ⊊ Rn is a closed unbounded convex set with BCEH then (i) E does not contain any affine real line, and (ii) for every affine line ℓ intersecting E in a ray and any hyperplane H transverse to ℓ, E is the epigraph of a convex function on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In particular, there are affine coordinates (x, y) on Rn in which E is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) for a convex function φ : Rn−1 → R+ satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) lim inf |x|→+∞ φ(x) |x| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) says that φ grows at least linearly at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We show in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='10 that linear growth is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (i) Assume that ℓ ⊂ E is an affine line and let us prove that E does not have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since E is a proper subset of Rn, there is a parallel translate ℓ′ of ℓ which is not contained in E, and hence ℓ′ \\E contains a ray L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let p be the endpoint of L, and let p′ ∈ L be an arbitrary other point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since E ∩L = ∅, there is a ball B around p′ such that Conv(B ∪{p})∩E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Clearly, there is a point q ∈ B such that the ray Lq with the endpoint p and containing q intersects the line ℓ, so the line segment from p to q belongs to Conv(E ∪ {p}) \\ E = h(E, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By moving p′ ∈ L to infinity we see that h(E, p) is unbounded, so E does not have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (ii) Since E is unbounded, it contains a ray L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Denote by ℓ the affine line containing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let ℓ′ be any parallel translate of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since E contains no affine lines by part (i), there is a point p ∈ ℓ′ \\E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The closed convex hull of the union of L and p contains the parallel translate L′ ⊂ ℓ′ of L passing through p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since E has BCEH, we conclude that L′ ⊂ Conv(E ∪ {p}) and L′ \\ E is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since E ∩ L′ is convex, L′ ∩ E is a closed ray with the endpoint on bE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This shows that E is a union of closed rays contained in parallel translates of the line ℓ, so it is an epigraph of a convex function defined on any hyperplane H ⊂ Rn transverse to ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Choosing H such that H ∩ E = ∅ there are affine coordinates (x, y) on Rn with H = {y = 0} and ℓ = {x = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In these coordinates, E is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) for a positive convex function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Finally, if condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) fails then there is a sequence (xk, yk) ∈ E with |xk| → +∞ and yk/|xk| → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The union of the line segments Lk connecting p = (0, −1) ∈ Rn−1×R to (xk, yk), intersected with the lower halfspace y ≤ 0, is then an unbounded subset of h(E, p) = Conv(E ∪ {p}) \\ E, contradicting the assumption that E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The growth condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) for an epigraph can always be achieved in suitable linear coordinates (even without the BCEH property) if there is a supporting hyperplane H ⊂ Rn for E such that the set E∩H is nonempty and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Indeed, we may then choose coordinates (x, y) on Rn such that H = {y = 0}, E ⊂ {y ≥ 0}, and 0 ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) fails, there is a sequence (xk, yk) ∈ E with |xk| → +∞ and yk/|xk| → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' After passing to a subsequence, a ray in E ∩ H lies in the closure of the union of the line segments Lk ⊂ E connecting the origin to (xk, yk), contradicting the assumption that the latter set is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If E is a closed convex set in Cn (n > 1) having BCEH then Cn \\ E is Oka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4 the set E does not contain any affine real line, and hence Cn \\ E is Oka by [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ 10 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Forstneriˇc The following lemma shows that the BCEH condition is stable under uniform approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume that φ : Rn−1 → R is a convex function whose epigraph Eφ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then for any ǫ > 0 and convex function ψ : Rn−1 → R satisfying |φ − ψ| < ǫ the epigraph Eψ also has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If Eψ fails to have BCEH then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 it has a boundary ray or an asymptote, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since dist(L, Eψ) = 0 and Eψ is convex, dist(x, Eψ) converges to zero as x ∈ L goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Thus, by making L shorter if necessary, we have that L ⊂ Eφ−2ǫ \\ Eφ+2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Hence, L lies out of Eφ+2ǫ but the vertical translation of L for 4ǫ pushes it in Eφ+2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since Eφ+2ǫ, being a translate of Eφ, has BCEH, this contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The contradiction shows that Eψ has BCEH as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ We now give a differential characterization of the BCEH property of an epigraph (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If φ : Rn−1 → R is a convex function of class C 1 satisfying condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2), then the epigraph E = {(x, y) ∈ Rn : y ≥ φ(x)} has BCEH if and only if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) lim |x|→∞ |x| � 1 − φ(x) x · ∇φ(x) � = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We first consider the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then, x is a single variable and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4) lim x→+∞ � x − φ(x) φ′(x) � = +∞ and lim x→−∞ � x − φ(x) φ′(x) � = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For every x ∈ R such that φ′(x) ̸= 0 the number (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5) ξ(x) = x − φ(x) φ′(x) is the first coordinate of the intersection of the tangent line to the graph of φ at the point (x, φ(x)) with the first coordinate axis y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) and convexity of φ we have that |φ′(x)| is bounded away from zero for all sufficiently big |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This shows that conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4) are invariant under translations, so we may assume that φ ≥ 0 and φ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It is easily seen that the function ξ is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If φ is of class C 2, we have that ξ′(x) = φ(x)φ′′(x)/φ′(x)2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume now that conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Pick a pair of sequences aj < bj in R with limj→∞ aj = −∞ and limj→∞ bj = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The intervals Ij = [ξ(aj), ξ(bj)] then increase to R as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We identify Ij with Ij × {0} ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since φ is convex, its epigraph lies above the tangent line at any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It follows that the set h(E, Ij) (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1)) is the bounded region in R×R+ whose boundary consists of Ij, the two line segments Lj and L′ j connecting the endpoints (ξ(aj), 0) and (ξ(bj), 0) of Ij to the respective points Aj = (aj, φ(aj)) and Bj = (bj, φ(bj)) on bE, and the graph of φ over [aj, bj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The supporting lines of Lj and L′ j intersect at a point Cj in the lower halfspace y < 0, and we obtain a closed triangle ∆j with the endpoints Aj, Bj, and Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Note that ∆j ∩ (R × {0}) = Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since φ grows at least linearly (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2)), the triangles ∆j ⊂ R2 exhaust R2 as j → ∞, and the set h(E, ∆j) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) is bounded for every j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Hence, E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This argument furthermore shows that for any point p = (0, −c) /∈ E there is a unique pair of tangent lines to bE passing through p such that, denoting by q1, q2 ∈ bE the respective points where these lines intersect bE, the convex hull Conv(E ∪ {p}) is the union of E and the triangle with vertices p, q1, q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Conversely, if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) fails then it is easily seen that E has a boundary ray or an asymptote, so it does not have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets 11 The case with n ≥ 3 now follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Pick a unit vector v ∈ Rn−1, |v| = 1, and let Lv denote the 2-plane in Rn passing through the origin and spanned by v and en = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then, Ev := E ∩ Lv = {(t, y) ∈ R2 : y ≥ φ(tv)} and the first condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4) reads (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6) lim t→+∞ � t − φ(tv) �n−1 j=1 vj ∂φ ∂xj (tv) � = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Writing x = tv with t ≥ 0 and v = x/|x|, this is clearly equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' As before, let p = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , 0, −c) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) holds then Conv(Ev ∪ {p}) ⊂ Lv is obtained by adding to Ev the triangle in Lv obtained by the two tangent lines to bEv passing through p as described in the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The sizes of these triangles are uniformly bounded with respect to the direction vector |v| = 1, and condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) implies that these triangles increase to Lv as c → +∞, uniformly with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since � |v|=1 Lv = Rn, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2 shows that Conv(E ∪ {p}) = � |v|=1 Conv(Ev ∪ {p}), and hence E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The converse is seen as in the special case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If φ : Rn−1 → R+ is a convex function of class C 1 such that lim |x|→+∞ x · ∇φ(x) |x| = +∞, then the epigraph E = {(x, y) ∈ Rn : y ≥ φ(x)} has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By restricting to planes as in the above proof, it suffices to consider the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We may assume that φ ≥ 0 and φ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since φ is convex, g(x) = φ′(x) is an increasing function and the above condition reads limx→±∞ g(x) = ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For any x0 > 0 and x ≥ x0 we have that ξ(x) := x − 1 g(x) � x 0 g(t)dt = � x 0 � 1 − g(t) g(x) � dt ≥ � x0 0 � 1 − g(t) g(x) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Letting x → +∞ we have that g(t) g(x) → 0 uniformly on t ∈ [0, x0], and hence the last integral converges to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Letting x0 → ∞ we see that limx→+∞ ξ(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The analogous argument applies when x → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Hence, conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) hold and therefore E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' There exist convex epigraphs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) having BCEH where the function φ grows linearly, although it cannot be too close to linear near infinity in the absence of boundary rays and asymptotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We give such an example in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let g : R → (−1, 1) be an odd, continuous, increasing function with limx→+∞ g(x) = 1 and � ∞ 0 (1−g(x))dx = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (An explicit example is g(x) = 2 πArctan x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') Its integral φ(x) = � x 0 g(t)dt for x ∈ R then clearly satisfies φ(x) ≥ 0, φ′(x) = g(x) ∈ (−1, +1) (hence φ grows linearly), and φ is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We now show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let x > 0 be large enough so that g(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We have that ξ(x) = x − 1 g(x) � x 0 g(t)dt = � x 0 � 1 − g(t) g(x) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Fix x0 > 0 and let x ≥ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then, ξ(x) ≥ � x0 0 (1 − g(t)/g(x))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since limx→+∞ g(x) = 1 and ξ is increasing for large enough |x|, it follows that limx→+∞ ξ(x) ≥ � x0 0 (1 − g(t))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Sending x0 → +∞ gives limx→+∞ ξ(x) ≥ � ∞ 0 (1 − g(t))dt = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Similarly we see that limx→−∞ ξ(x) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) holds, and hence the epigraph of φ has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By using the idea in the above example we now prove the following approximation result, which extends Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 to a much bigger class of convex epigraphs (see Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 12 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Forstneriˇc Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume that φ : Rn−1 → R+ is a convex function such that the set {φ = 0} is nonempty and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Given numbers ǫ > 0 (small) and R > 0 (big) there is a smooth convex function ψ : Rn−1 → R such that ψ < φ on Rn−1, φ(x) − ψ(x) < ǫ for all |x| ≤ R, and the epigraph Eψ = {y ≥ ψ} has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5 the function φ grows at least linearly near infinity (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='7) A = lim inf |x|→∞ φ(x) |x| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since the set φ = 0 does not contain any affine line, Azagra’s result [6, Theorem 1 and Proposition 1] implies that for every ǫ > 0 there is a smooth strictly convex function ψ on Rn−1 satisfying φ − ǫ < ψ < φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Replacing φ by ψ − minx ψ(x) ≥ 0 we may therefore assume that φ is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By increasing the number R > 0 if necessary, we may assume that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='8) φ(x) |x| ≥ A 2 for all |x| ≥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Pick a number r ∈ (0, 1) close to 1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='9) (1 − r) sup |x|≤R φ(x) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Choose a smooth increasing function h : R → R+ such that h(t) = 0 for t ≤ R, lim t→+∞ h(t) = 1, and � ∞ 0 (1 − h(t))dt = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (We can take a smoothing of the Arctan function used in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') Set H(x) = � |x| 0 h(s)ds for x ∈ Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Clearly, H ≥ 0 is a radially symmetric smooth convex function that vanishes on |x| ≤ R and satisfies H(x) ≤ |x| for all x ∈ Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' With A and r as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='9) we set δ = A(1 − r) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We claim that the function ψ(x) = rφ(x) + δH(x) for x ∈ Rn−1 satisfies the conditions in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Clearly, ψ ≥ rφ is a smooth convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For |x| ≤ R we have H(x) = 0, so ψ(x) = rφ(x) ≤ φ(x) and φ(x) − ψ(x) = (1 − r)φ(x) < ǫ by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If |x| > R then φ(x)/|x| ≥ A/2 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='8) and H(x) < |x|, which implies ψ(x) |x| ≤ rφ(x) |x| + δ ≤ φ(x) |x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Indeed, we have that φ(x) |x| − r φ(x) |x| = (1 − r)φ(x) |x| ≥ A(1−r) 2 = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Hence, ψ ≤ φ on Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' It remains to show that the epigraph Eψ satisfies BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We shall verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3), which is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6) with uniform convergence with respect to the vector v = x/|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Write gv(t) = r∂φ(tv) ∂t , k(t) = δh(t), ˜gv(t) = ∂ψ(tv) ∂t = gv(t) + k(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets 13 The quantity in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6) associated to the function ψ is given by ξv(t) = t − ψ(tv) ˜gv(t) = � t 0 � 1 − gv(s) + k(s) gv(t) + k(t) � ds = � t 0 gv(t) − gv(s) gv(t) + k(t) ds + � t 0 k(t) − k(s) gv(t) + k(t)ds ≥ � t 0 gv(t) − gv(s) gv(t) + δ ds + � t 0 k(t) − k(s) gv(t) + δ ds, where the last inequality holds since the functions gv and k are nonnegative and increasing and k < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Pick c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We will show that for large enough t > 0 and any unit vector v ∈ Rn−1 the above expression is bigger than or equal to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Choose positive numbers t0, a, t1 as follows: t0 = 3c, a = max{3 max |v|=1 gv(t0), 3δ}, � t1 0 (k(t1) − k(s))ds > ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Such t1 exists since limt→+∞ � t 0(k(t) − k(s))ds = δ � ∞ 0 (1 − h(s))ds = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since the integrands in the bound for ξv(t) are nonnegative, we have for t ≥ max{t0, t1} and |v| = 1 that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='10) ξv(t) ≥ � t0 0 gv(t) − gv(s) gv(t) + δ ds + � t1 0 k(t) − k(s) gv(t) + δ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume that for some such (t, v) we have that gv(t) + δ ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since a ≥ 3δ, it follows that gv(t) ≥ 2δ and hence gv(t) gv(t) + δ ≥ 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Furthermore, from a ≥ 3 max|v|=1 gv(t0) we get for 0 ≤ s ≤ t0 that gv(s) gv(t) + δ ≤ gv(t0) a ≤ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' These two inequalities imply that the first integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='10) is bounded below by t0/3 ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If on the other hand gv(t) + δ < a then the denominator of the second integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='10) is at most a, so the integral is ≥ c by the choice of t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This shows that ξv(t) ≥ c for all |v| = 1 and t ≥ max{t0, t1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since c was arbitrary, condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) holds and hence Eψ has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ The following observation will be used in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Denote by B the open unit ball in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let Eφ ⊂ Rn be a closed convex set of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) with C 1 boundary having BCEH, where the function φ : Rn−1 → R is bounded from below and strictly convex near infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then there is an r0 > 0 such that for every r ≥ r0 the convex hull Conv(Eφ ∪ rB) = {y ≥ ψ(x)} is a closed convex set with BCEH, and ψ : Rn−1 → R is a convex function of class C 1 such that ψ ≤ φ and these functions agree near infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Furthermore, if r ≥ r0 is large enough then the function φt : Rn−1 → R defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='11) φt(x) = (1 − t)φ(x) + tψ(x), x ∈ Rn−1 is strictly convex for every t ∈ (0, 1), and for any 0 < t0 < t1 < 1 the closure of the set {(x, y) ∈ Rn : φt1(x) < y < φt0(x)} is a strictly convex cap with the base in the strictly convex hypersurface {y = φt0(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Forstneriˇc Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Consider the function on Rn−1 given by ˜φr(x) = � min{φ(x), − � r2 − |x|2}, |x| < r, φ(x), |x| ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (Note that ˜φr may be discontinuous at the points of the sphere |x| = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') The convex hull of its epigraph E˜φr equals Conv(E∪rB), which is closed by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 (iv), and the set h(E, rB) = Conv(E ∪ rB) \\ E is bounded since E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By smoothing ˜φr we get a function ˜ψr of class C 1 which agrees with φ near infinity such that Conv(E ˜ψr) = Conv(E ∪ rB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By [28, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2] we conclude that Conv(E ∪ rB) has C 1 boundary, so it is the epigraph Eψr of a convex function ψr : Rn−1 → R of class C 1 which agrees with φ near infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since φ grows at least linearly, there is a function τ(r) defined for r ∈ R+ large enough such that ψr(x) = − � r2 − |x|2 for |x| ≤ τ(r) and τ(r) → +∞ as r → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By choosing r large enough, the compact set of points where the function φ fails to be strictly convex is contained in the ball |x| < τ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since on this ball we have that ψr(x) = − � r2 − |x|2 which is strictly convex, the convex combinations φt in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='11) of φ and ψ = ψr are strictly convex on Rn−1 for all 0 < t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For such r, the last statement in the proposition is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (Note that the strictly convex functions ρt(x, y) = exp(ψt(x) − y) − 1 for t ∈ (0, 1) correspond to those used in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 For the definition and the main theorem on Oka manifolds, see [20, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We shall use the following version of the Oka principle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' see [22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assume that X is a Stein manifold, K is a compact O(X)-convex set in X, X′ is a closed complex subvariety of X, Ω is an Oka domain in a complex manifold Y , f : X → Y is a continuous map which is holomorphic on a neighbourhood of K, f|X′ : X′ → Y is holomorphic, and f(X \\ ˚ K) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then there is a homotopy {ft}t∈[0,1] of continuous maps ft : X → Y connecting f = f0 to a holomorphic map f1 : X → Y such that for every t ∈ [0, 1] the map ft is holomorphic on a neighbourhood of K, it agrees with f on X′, it approximates f uniformly on K and uniformly in t ∈ [0, 1] as closely as desired, and ft(X \\ ˚ K) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4 there are complex coordinates z = (z′, zn) on Cn such that the given set E is an epigraph of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We shall write z = (x, y) where x = (z′, ℜzn) ∈ Cn−1 × R ∼= R2n−1 and y = ℑzn ∈ R, so E = Eφ = {y ≥ φ(x)} where φ ≥ 0 is a convex function as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let the set K ⊂ X and the map f0 : K → Cn be as in the theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' in particular, f0(bK) �� Cn \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Thus, there are an open neighbourhood U ⊂ X of K and ǫ > 0 such that f0 is holomorphic in U and f0(U \\ ˚ K) ⊂ Cn \\ Eφ−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By Azagra [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='8] there is a a real analytic strictly convex function φ0 : R2n−1 → R such that φ − ǫ < φ0 < φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Its epigraph E0 = {(x, y) ∈ Cn : y ≥ φ0(x)} is a closed strictly convex set with real analytic boundary which has BCEH by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='7, and f0(U \\ ˚ K) ⊂ Cn \\ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let B denote the open unit ball in Cn centred at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Recall the notation h(E, K) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Pick a number r0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We can find an increasing sequence rk > 0 diverging to infinity such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) h(E0, rkB) ⊂ rk+1B for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets 15 Indeed, since E0 has BCEH, the set h(E0, rkB) is bounded for each k, and hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) holds if the number rk+1 is chosen large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Set Ek+1 = Conv(E0 ∪ rkB) = E0 ∪ h(E0, rkB) for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='. We clearly have that E0 ⊂ E1 ⊂ · · · ⊂ �∞ k=0 Ek = Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Furthermore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) shows that for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , k + 1 we have that E0 ⊂ Ej ⊂ E0 ∪ rk+1B and hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2) Ek+2 = Conv(Ej ∪ rk+1B) for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='12 shows that for each k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' we have Ek = {y ≥ φk(x)} where φk a convex function of class C 1 which agrees with φ0 near infinity, and Ek has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Hence, Ωk = Cn \\ Ek = {(x, y) ∈ Cn : y < φk(x)} is an Oka domain for every k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In view of Ek+2 = Conv(Ek∪rk+1B) (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2)), Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='12 also shows that if rk+1 is chosen large enough then the function (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3) ψt = (1 − t)φk + tφk+2 : Cn−1 × R → R is strictly convex for every t ∈ (0, 1), and for each 0 < t0 < t1 < 1 the closure of the set (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4) C = {(x, y) : ψt1 < y < ψt0} is a strictly convex cap as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (Note that the strictly convex functions ρt(x, y) = exp(ψt(x) − y) − 1 for t ∈ (0, 1) correspond to those used in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') Choose an exhaustion D0 ⊂ D1 ⊂ · · · ⊂ �∞ k=0 Dk = X by smoothly bounded, relatively compact, strongly pseudoconvex domains with O(X)-convex closures such that K ⊂ D0 ⊂ ¯D0 ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For consistency of notation we set D−1 = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We now construct a sequence of holomorphic maps fk : ¯Dk → Cn satisfying the following conditions for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' : (a) fk(Dk \\ Dk−1) ⊂ Ωk = Cn \\ Ek, (b) fk+1(Dk \\ Dk−1) ⊂ Ωk, and (c) fk+1 approximates fk uniformly on ¯Dk−1 as closely as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' For k = 0 the initial map f0 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 satisfies condition (a) while conditions (b) and (c) are void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assuming inductively that we found maps f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' , fk satisfying these conditions, the construction of the next map fk+1 is made in two steps as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By compactness of the set fk(bDk) ⊂ Ωk = {y < φk(x)} we can choose t0 ∈ (0, 1) small enough such that f(bDk) ⊂ {y < ψt0(x)}, where the function ψt (t ∈ [0, 1]) is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) we can also choose t1 ∈ (t0, 1) sufficiently close to 1 such that Ek+1 ⊂ {(x, y) : y ≥ ψt1(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 applied to the map fk : ¯Dk → Cn, the set Ek, and the strictly convex cap C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4) (which corresponds to C1 in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1) gives holomorphic map gk : ¯Dk → Cn approximating fk on Dk−1 and satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5) gk(bDk) ⊂ {(x, y) : y < ψt1(x)} ⊂ Cn \\ Ek+1 = Ωk+1 and gk(Dk \\ Dk−1) ⊂ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In the second step, we use that Ωk+1 is an Oka domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since Ωk+1 is contractible and gk(bDk) ⊂ Ωk+1 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='5), gk extends from ¯Dk to a continuous map X → Cn sending X \\ Dk to Ωk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 applied to gk gives a holomorphic map fk+1 : ¯Dk+1 → Cn approximating gk on ¯Dk and satisfying fk+1(Dk+1 \\ Dk) ⊂ Ωk+1 (which is condition (a) for k + 1) and fk+1(Dk \\ Dk−1) ⊂ Ωk (condition (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Since fk+1 approximates gk on ¯Dk and gk approximates fk on ¯Dk−1, fk+1 also satisfies condition (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This completes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' 16 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Forstneriˇc If the approximations are close enough then the sequence fk converges uniformly on compacts in X to a holomorphic f : X → Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Conditions (a)–(c) and the fact that the sets Ek exhaust Cn imply that f is a proper holomorphic map satisfying f(X \\ ˚ K) ⊂ Ω0 = Cn \\E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' To construct proper holomorphic immersions and embeddings in suitable dimensions given in the theorem, we use the general position argument at every step to ensure that every map fk in the sequence is an immersion or an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' [20, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=') If the convergence is fast enough then the same holds for the limit map f by a standard argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Given a holomorphic map f0 : K → Cn with f0(bK) ⊂ Cn \\ Eφ as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='11 furnishes a closed convex set Eψ ⊃ Eφ with BCEH such that f0(bK) ⊂ Cn \\ Eψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 with Eψ gives the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ We have the following analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3 with interpolation on a closed complex subvariety of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Unlike in the above corollary, approximation of E from the outside by convex sets enjoying BCEH cannot be used since the subvariety f(X′) may have zero distance to bE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This results extends the case of [24, Theorem 15] when E is a compact convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let E be a closed convex set in Cn (n > 1) with C 1 boundary which is strictly convex near infinity and has bounded convex exhaustion hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Let X be a Stein manifold, K ⊂ X be a compact O(X)-convex set, U ⊂ X be an open set containing K, X′ be a closed complex subvariety of X, and f0 : U ∪ X′ → Cn be a holomorphic map such that f0|X′ : X′ → Cn is proper holomorphic and f0(bK ∪ (X′ \\ K)) ∩ E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Given ǫ > 0 there exists a proper holomorphic map f : X → Cn satisfying the following conditions: (a) f(X \\ ˚ K) ⊂ Cn \\ E, (b) ∥f − f0∥K < ǫ, (c) f|X′ = f0|X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' If 2 dim X ≤ n then f can be chosen an immersion (and an embedding if 2 dim X + 1 ≤ n) provided that f0|X′ is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This is proved by a small modification of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3, similar to the one in [24, proof of Theorem 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The initial step in the proof, approximating E from the outside by a strictly convex set, is unnecessary since bE is strictly convex near infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The main (and essentially the only) change comes in the choice of the exhaustion Dk of the Stein manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' In the inductive step when constructing the map fk+1, we must assume in addition that fk(bDk ∩ X′) ⊂ Ωk+1 = Cn \\Ek+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Then, we push the image of bDk out of Ek+1 by the same method as before, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='1 but ensuring that the modifications are kept fixed on X′ and small near bDk ∩ X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This is possible since the method from [15] is applied locally near bDk (away from bDk ∩ X′), and these local modifications are glued together by preserving the value of the map on X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We refer to [24, proof of Theorem 15] for a more precise description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This gives the next holomorphic map fk+1 : X → Cn satisfying fk+1(X \\ Dk) ⊂ Ωk+1, fk+1|X′ = fk|X′, and conditions (b) and (c) in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' We then choose the next domain Dk+1 ⊂ X big enough such that fk+1(bDk+1 ∩X′) ⊂ Ωk+2 = Cn \\Ek+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' This is possible since the map fk+1|X′ = f0|X′ : X′ → Cn is proper, f0(X′ \\ ˚ K) ⊂ Ω = Cn \\ E, and the domain Ωk+2 agrees with Ω near infinity by the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Clearly the induction step is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Assuming that the approximations are close enough, the sequence fk converges to a limit holomorphic map f : X → Cn satisfying the stated conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' □ Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The first named author is supported by grants P1-0291, J1-3005, and N1- 0137 from ARRS, Republic of Slovenia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The second named author is supported by the European Union (ERC Advanced grant HPDR, 101053085) and grants P1-0291, J1-3005, and N1-0237 Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets 17 from ARRS, Republic of Slovenia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' The authors wish to thank Antonio Alarc´on for helpful discussions and information concerning the case pertaining to minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} 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+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=', 307(3):381– 399, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' [39] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Stensønes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Proper holomorphic mappings from strongly pseudoconvex domains in C2 to the unit polydisc in C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Scand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=', 65(1):129–139, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' Barbara Drinovec Drnovˇsek Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content=' e-mail: barbara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='drinovec@fmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='uni-lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='si Franc Forstneriˇc Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia e-mail: franc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='forstneric@fmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='uni-lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'} +page_content='si' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}