On the convexity of the weakly compact Chebyshev sets in Banach spaces
type: Item
instance of: scholarly article
title: On the convexity of the weakly compact Chebyshev sets in Banach spaces
zbMATH Open document ID: 0955.46005
DOI: 10.1007/BF02773563
author: Vassilis Kanellopoulos
published in: Israel Journal of Mathematics
publication date: 2001-02-14T00:00:00Z
review text: A subset \(M\) of a normed linear space \(X\) is called a Chebyshev set if to each point \(x\) of \(X\) there exists a unique point of \(M\) nearest to \(x\). The main result of this paper is a sufficient condition (almost smooth) for a Banach spaces \(X\) that every weakly compact Chebyshev set is convex. As an immediate consequence a former result of A. Brøndsted and A. L. Brown is partially extended and a known result in Hilbert spaces is proved in a different way. An important tool is to generalize the notion of extreme points to so-called \(n\)-extreme points. The paper includes all of the proofs.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B07
zbMATH DE Number: 1455687
zbMATH Keywords: Chebyshev sets
zbMATH Keywords: extreme points
zbMATH Keywords: almost smooth
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773563
OpenAlex ID: W2090653176
description: scientific article; zbMATH DE number 1455687