On approximation properties of sets with convex complement
type: Item
instance of: scholarly article
title: On approximation properties of sets with convex complement
zbMATH Open document ID: 0857.41030
DOI: 10.1007/BF02309389
author: V. S. Balaganskij
published in: Mathematical Notes
publication date: 1996-02-21T00:00:00Z
review text: Let \(X\) be a Banach space and \(M\) a non-empty closed subset of \(X\). If \(x\in X\) then \(\phi(x)\) denotes the distance of \(x\) from \(M\) and \(P_M(x)\) denotes the set of points \(z\in M\) such that \(|x-z|= \phi(x)\). One denotes by \(\text{wAC}(M)\) the set of points \(x\in X\) of `weak approximate compactness' relative to \(M\) and by \({\mathcal F}_{CC}\) the family of non-empty closed subsets of \(X\) which have non-empty convex complements. If \(M\in {\mathcal F}_{CC}\) then \(-\phi\) is a convex function and if, further, at a point \(x\in X\backslash M\) the function \(\phi\) is Gâteaux differentiable with Gâteaux derivative \(\phi'(x)\in X^*\) then the hyperplane \(N=\{z\in X:\phi'(x)(z)= \phi'(x)(z)-\phi(x)\}\) is a subset of \(M\); \(\phi(x)\) is equal to the distance of \(x\) from \(N\) if and only if \(|\phi'(x)|=1\); and \(P_M(x)\) is non-empty if and only if \(|\phi'(x)|=1\) and \(\phi'(x)\) attains its bound. The paper explores this situation. A number of similar results are obtained of which the following is representative. If \(X\) is a Banach space then the following statements are equivalent: (1) \(X\) is reflexive, (2) for each \(M\in {\mathcal F}_{CC}\) the set \(E_M=\{x: P_M(x)\neq \emptyset\}\) is dense in \(X\), (3) for each \(M\in{\mathcal F}_{CC}\) the set \(X\backslash \text{wAC}(M)\) is of first category. It is further shown that if \(X\) is a Banach space and \(\eta\in [0,1)\) then \(\dim X=\infty\) if and only if there exist \(M\in {\mathcal F}_{CC}\) and a point \(x\in X\backslash M\) at which the Gâteaux derivative \(\phi'(x)\) exists and \(|\phi'(x)|=\eta\).
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46G05
zbMATH DE Number: 846455
zbMATH Keywords: distance function
zbMATH Keywords: Gâteaux differentiability
reviewed by: Aldric L. Brown
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 846455