A Chebyshev set and its distance function
type: Item
instance of: scholarly article
title: A Chebyshev set and its distance function
zbMATH Open document ID: 1032.46023
published in: Journal of Approximation Theory
publication date: 2003-04-02T00:00:00Z
review text: Let \(K\) be a Chebyshev subset of a smooth Banach space \(B\). An old open question in Banach space geometry asks if \(K\) must be convex. The answer is not known even if \(B\) is a Hilbert space. There are many beautiful partial results. For example, compact Chebyshev subsets of a smooth Banach space are convex. Other partial results involve conditions on \(B\), and its dual space, \(B^*\), or geometric or topological properties of \(K\). This paper shows that \(K\) is convex if the distance function, \(d(x)=\min\{\|x-k\|: k \in K\}\), satisfies smoothness conditions such as Gâteaux differentiability or continuity. When stricter smoothness conditions are assumed for \(B\) or \(B^*\), less restrictive conditions are required for \(d\). The paper contains, for example, the following nice statement: Theorem. If the norms on \(B\) and \(B^*\) are Fréchet differentiable, then \(K\) is convex if and only if \(d\) is Fréchet differentiable.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 1891512
zbMATH Keywords: convex sets
zbMATH Keywords: uniformly smooth spaces
zbMATH Keywords: Fréchet derivatives
zbMATH Keywords: Chebyshev subset
author: Zili Wu
reviewed by: Daniel Wulbert
MaRDI profile type: Publication
full work available at URL: jath.2002.3728
OpenAlex ID: W2010612323
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DOI: 10.1006/JATH.2002.3728
description: scientific article; zbMATH DE number 1891512