Chebyshev centers and some geometric properties of Banach spaces
type: Item
instance of: scholarly article
title: Chebyshev centers and some geometric properties of Banach spaces
zbMATH Open document ID: 1367.46015
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-01-17T00:00:00Z
review text: Let \(X\) be a real Banach space, \(\emptyset\neq V\subset X\) closed and \(\emptyset\neq F\subset X\) bounded. For \(v\in V\) and \(\delta >0\) put \(r(v,F)=\sup\{\|v-x\| : x\in F\}\), \(\mathrm{rad}_V(F)=\inf\{r(v,F) : v\in V\}\) (the Chebyshev radius of \(F\) with respect to \(V\)), \(\mathrm{cent}_V(F)=\{v\in V : v(v,F)= \mathrm{rad}_V(F)\}\) (the Chebyshev center of \(F\) with respect to \(V\)) and \(\delta\)-\(\mathrm{cent}_V(F)=\{v\in V : v(v,F)\leq \mathrm{rad}_V(F)+\delta\}\). Denote also by \(CL(X), CB(X), CC(X)\) and \(K(X)\) the family of all nonempty closed subsets of \(X\), nonempty closed bounded subsets of \(X\), nonempty closed convex subsets of \(X\), and nonempty compact subsets of \(X\), respectively. Let \(V\in CL(X)\) and \(\mathcal F\subset CB(X)\) be such that \(\mathrm{cent}_V(F)\neq\emptyset\) for every \(F\in\mathcal F\). One says that the pair \((V,\mathcal F)\) has the property: {\parindent=9mm \begin{itemize}\item[(P1)] if, for every \(F\in\mathcal F\) and \(\varepsilon >0\), there exists \(\delta>0\) such that \(\delta\)-\(\mathrm{cent}_V(F)\subset \mathrm{cent}_V(F)+\varepsilon B_X\); \item[(P2)] if, for every \(\varepsilon >0\), there exists \(\delta>0\) such that \(\delta\)-\(\mathrm{cent}_V(F)\subset \mathrm{cent}_V(F)+\varepsilon B_X\) for all \(F\in\mathcal F\). \end{itemize}} The properties (P1) and (P2) were introduced by \textit{J. Mach} [J. Approx. Theory 29, 223--230 (1980; Zbl 0467.41015)] in his study of the continuity properties of the Chebyshev center map. The main aim of the paper is the study of properties (P1) and (P2) in connection with some geometric properties of Banach spaces. For example: \(X\) is reflexive with the Kadets-Klee property iff the pair \((V,K(X))\) has property (P1) for every \(V\in CC(X)\) (Theorem 2.3). If \(X\) is compactly locally uniformly convex (CLUR), then, for fixed \(V\in CC(X)\), the pair \((V,\mathcal F)\) has property (P1), where \(\mathcal F=\{F\in K(X) : \mathrm{cent}_V(F)\neq\emptyse…
reviewed by: Costică Mustăţa
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A46
zbMATH DE Number: 6675628
zbMATH Keywords: Chebyshev center
zbMATH Keywords: Banach space
zbMATH Keywords: Kadets-Klee property
zbMATH Keywords: Hausdorff metric
MaRDI profile type: Publication
full work available at URL: j.jmaa.2016.12.026
OpenAlex ID: W2567226646
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DOI: 10.1016/J.JMAA.2016.12.026
description: scientific article; zbMATH DE number 6675628