Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

For a proper holomorphic map F from a ball B n into another ball B N , we give a criterion when a rational map F is equivalent to a polynomial one.As application, we show that proper rational holomorphic maps from B 2 into B N of degree two are equivalent to polynomial maps, and we also give an example of a rational holomorphic maps of degree 3 that are 'almost' linear but are not equivalent to holomorphic polynomial maps.

We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.

Denote by \(\triangle\) the open unit disc in \({\Bbb C}\). We prove that given a discrete subset S of a connected Stein manifold M there is a proper holomorphic map \(f:\triangle\to M\) such that \(S\subset f(\triangle)\); if \(\dim M\ge 3\) the map f can be chosen to be an embedding. In addition we prove that we can prescribe higher order contacts of \(f(\triangle)\) with given one dimensional submanifolds in M.

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

Applications holomorphes propres d'un domaine quasi-disqué sur un domaine disqué

A General Hopf Lemma and Proper Holomorphic Mappings between Convex Domains in ℂ n

We study the convergence of an iterative projection/reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics. There are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem: nonconvexity and infeasibility. The algorithm that we developed, called relaxed averaged alternating reflections (RAAR), was designed primarily to address infeasibility, though our strategy has advantages for nonconvex problems as well. In the present work we investigate the asymptotic behavior of the RAAR algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a Hilbert space with empty intersection, and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox-regular set, also possibly nonintersecting. The nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection. To place the RAAR algorithm in context, we develop parallel statements about the standard alternating projections algorithm and gradient descent. All of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings.

We construct a functorial pushforward homomorphism in geometric Hodge filtered complex cobordism along proper holomorphic maps between arbitrary complex manifolds. This significantly improves previous results on such transfer maps and is a much stronger result than the ones known for differential cobordism of smooth manifolds. This enables us to define and provide a concrete geometric description of Hodge filtered fundamental classes for all proper holomorphic maps. Moreover, we give a geometric description of a cobordism analog of the Abel–Jacobi invariant for nullbordant maps which is mapped to the classical invariant under the Hodge filtered Thom morphism. For the latter we provide a new construction in terms of geometric cycles.

In this paper, we establish some properties of the multivalued mapping (x, d) ⇒ DC (x; d) that associates to every element x of a linear normed space X the set of linear continuous functionals of norm d ≥ 0 and which separates the closed ball B (x; d) from a closed convex set C ⊂ X. Using this mapping we give links with other important concepts in convex analysis (ε-approximation element, ε-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of ε-approximation elements with respect to a nonvoid closed convex set as a generalization of a known result of Garkavi. Also, we give some properties of univocity and monotonicity of mapping DC. Mathematics Subject Classification (2010): 32A70, 41A65, 46B20, 46N10. Received 18 May 2023; Accepted 27 November 2023

Introduction. This paper gives counterexamples for two unrelated conjectures which pertain to convexity theory. Supposef is any function defined on a closed convex set C in finitedimensional Euclidean space. Its envelope function env f is defined as the pointwise supremum of all linear functions which f dominates everywhere. It is elementary and well known that env f is convex, and hence continuous in the relative interior of C. It is similarly known that env f is lower-semicontinuous on the boundary of C. Witsenhausen [14] uses and studies envelope functions. He suggested the following orally: CONJECTURE 1. Iff is continuous everywhere on C, then envf is also continuous everywhere on C. I provide a very simple counterexample in 3-space which shows that this is not true. Incidentally, though the proof is not on paper, I feel fairly certain that Witsenhausen's conjecture is correct in 2-space. If C is a closed convex set in finite-dimensional Euclidean space, then for any point x in the space there is a unique point P(x) in C which is nearest to x. The mapping P is called the projection onto C. (Many interesting results about this mapping are given in Phelps [9], [10].) It is easy to prove and well known that P is a contraction mapping, hence continuous. It is also easy to see that P need not have much in the way of differentiability properties, for if C is a polyhedron in 2-space, then it is easy to find points and directions at which P does not even have a two-sided directional derivative. The existence in general of a one-sided directional derivative, however, is not so easy to decide. This question does not seem to be raised in such natural places as Eggleston [4] and Valentine [11]. CONJECTURE 2. For all points and directions, P has a one-sided directional derivative. My mildly complicated counterexample is a set C (in 3-space), which is the convex hull of a countable infinity of points. I suspect that the conjecture is correct in 2-space. In n-space it would be interesting to know whether the set of points and directions at which the conjecture holds is dense in EnXSn-. If not, the convex set would have to be extremely jagged!

We develop a theory of simultaneous metric projection in a normed linear space $X$ and present various characterizations of simultaneous metric projection onto closed convex sets in terms of the elements of $X^{\ast}$. Also, we characterize the elements of simultaneous metric projection onto closed convex sets in terms of extreme points of the closed unit ball $B_{X^{\ast}}.$ Finally, as an application, we give various characterizations of simultaneous metric projection onto subspaces of the Banach space $C(Q,Y).$

Let S be a 3-convex subset of the plane.If (cl S ~ £>) S int (cl S) or if (cl S ~ S) g bdry (cl S), then S is expressible as a union of four or fewer convex sets.Otherwise, S is a union of six or fewer convex sets.In each case, the bound is best possible.1* Introduction.Let S be a subset of R d .Then S is said to be 3-convex iff for every three distinct points in S, at least one of the segments determined by these points lies in S. Valentine [2] has proved that for S a closed, 3-convex subset of the plane, S is expressible as a union of three or fewer closed convex sets.We are interested in obtaining a similar decomposition without requiring the set S to be closed.The following definitions and results obtained by Valentine will be useful.

We develop a theory of best simultaneous approximation for closed convex sets in a conditionally complete lattice Banach space X with a strong unit. We study best simultaneous approximation in X by elements of closed convex sets, and give necessary and sufficient conditions for the uniqueness of best simultaneous approximation. We give a characterization of simultaneous pseudo-Chebyshev and quasi-Chebyshev closed convex sets in X. Also, we present various characterizations of best simultaneous approximation of elements by closed convex sets in terms of the extremal points of the closed unit ball B X* of X*.

Let Gi be closed Lie groups of U(n), Ωi be bounded Gi-invariant domains in ℂn which contains 0, and O(ℂn)Gi = ℂ, for i = 1, 2. It is known that if f: Ω1 →} Ω2 is a proper holomorphic mapping, and f−1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f.

This chapter develops basic properties of complex Euclidean space. Some of the main ideas are unitary transformations, the holomorphic automorphism group of the unit ball, the use of Hermitian forms, and proper holomorphic mappings. We also gather some elementary combinatorial information.