Ambiguous loci of mutually nearest and mutually furthest points in Banach spaces
type: Item
instance of: scholarly article
title: Ambiguous loci of mutually nearest and mutually furthest points in Banach spaces
zbMATH Open document ID: 1052.41019
author: C. Li
author: Hong-Kun Xu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-09-23T00:00:00Z
review text: This paper deals with the nature (in the topological sense) of the set where certain minimization or maximization problems are not well-posed. We recall that these problems are well-posed if there is a unique solution and every minimizing (maximizing) sequence converges strongly to the solution. Let \(X\) be a separable strictly convex Banach space. Let \({\mathcal K}(X)\) denote the family of non-empty boundedly compact convex sets equipped with the \(H_{\rho}\)-topology. Let \({\mathcal K}_{G}(X)\) denote the closure of the family of sets that are disjoint from \(G\). Let \({\mathcal V}(G) = \{A \in {\mathcal K}_{G}(X) : \min(A,G)\) is not well-posed\}. The main result of this paper shows that the collection of nonempty closed sets \(G\) for which \({\mathcal V}(G)\) is everywhere uncountable, is residual in the collection of all nonempty closed sets. Similar result is obtained in the case of farthest points by considering the Hausdorff distance and sets are assumed in addition to be bounded/compact.
reviewed by: T.S.S.R.K. Rao
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 2102590
zbMATH Keywords: minimization problem
zbMATH Keywords: well-posed
MaRDI profile type: Publication
full work available at URL: j.na.2004.04.008
OpenAlex ID: W2117708628
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DOI: 10.1016/J.NA.2004.04.008
description: scientific article; zbMATH DE number 2102590