A note on ''Approximation of bounded sets''
type: Item
instance of: scholarly article
title: A note on ''Approximation of bounded sets''
zbMATH Open document ID: 0603.41025
DOI: 10.1016/0021-9045(85)90072-3
author: Dan Amir
published in: Journal of Approximation Theory
publication date: 1985-01-01T00:00:00Z
review text: Let X be a normed linear space, V and F non-empty subsets of X with V convex and F bounded. An element \(v_ 0\in V\) is called best simultaneous approximation (b.s.a.) to F if \(\sup_{x\in F}\| x-v_ 0\| =\inf_{v\in V}\sup_{x\in F}\| x-v\|.\) Let K be a \(\sigma (V,X^*)\)-compact subset of the dual unit ball \(B(X^*)\) which norms F- V i.e. such that \(\| x-v\| =\sup_{f\in K}f(x-v)\) for all \(x\in F\), \(v\in V\). On the compact space (K,\(\sigma\) (V,K)), consider the upper semi-continuous function \[ U^+_ F(f)=\limsup_{K\ni g\to \sigma (V,K)}f\sup_{x\in F}g(x)=\inf \sup_{g\in W}\sup_{x\in F}g(x) \] over all \(\sigma\) (V,K)-neighbourhoods W of f in K. The following characterization of b.s.a. (Chebyshev centre) was given by \textit{J. H. Freilich} and \textit{H. W. McLaughlin} in the paper cited in the title [ibid. 34, 146-158 (1982; Zbl 0504.41013)] - Theorem 2: Let \(v_ 0\in V\) and \(E=\sup_{f\in F}\| f-v_ 0\|.\) Then \(v_ 0\) is a b.s.a. if and only if for each \(v\in V\) there exists an \(L\in ext(K)\) such that (i) \(U^+_ F(L)-L(v_ 0)=E,\) and (ii) \(L(v-v_ 0)\leq 0.\) By showing that the necessity part of this theorem is false, the following correct version is given in the paper under review: Theorem. \(v_ 0\in V\) is a b.s.a. of F in V if and only if for every \(v\in V\) there is an L in the \(\sigma\) (V,K)-closure of ext(K) which satisfies (i) and (ii) above.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A28
zbMATH DE Number: 3973504
zbMATH Keywords: best simultaneous approximation
zbMATH Keywords: Chebyshev centre
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: 0021-9045(85)90072-3
OpenAlex ID: W2014175944
cites work: Chebyshev centers in normed spaces
cites work: Approximation of bounded sets
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description: scientific article; zbMATH DE number 3973504