On well-posed mutually nearest and mutually furthest point problems in Banach spaces
type: Item
instance of: scholarly article
title: On well-posed mutually nearest and mutually furthest point problems in Banach spaces
zbMATH Open document ID: 1069.46010
author: C. Li
author: Renxing Ni
publication date: 2004-11-26T00:00:00Z
review text: For a Banach space \(X\), denote by \(\mathcal B(X)\), \(\mathcal C(X)\), \(\mathcal K(X)\) the spaces of all nonempty closed bounded, nonempty closed bounded convex, respectively, nonempty compact convex subsets of \(X\). Equipped with the Hausdorff metric \(h\), \(\mathcal B(X)\) is a complete metric space and \(\mathcal C(X)\), \(\mathcal K(X)\) are closed subset of \(\mathcal B(X),\) hence complete, too. For a nonempty closed subset \(G\) of \(X\), denote by \(\mathcal K_G(X)\) the closure with respect to \(h\) of the set \(\{A\in \mathcal C(X) : A\cap G = \emptyset\}.\) The paper is concerned with the well-posedness of the problem \(\min(A,G)\) (resp. \(\max(A,G)\)) of finding \((x_0,z_0)\in A\times G\) such that \(\| z_0-x_0\| = \lambda_{AG}:= \inf\{\| z-x\| : x\in A\), \(z\in G\}\) (resp. \( \| z_0-x_0\| = \mu_{AG}:= \sup\{\| z-x\| : x\in A\), \(z\in G\}\)). The authors prove that if \(X\) is a strictly convex Banach space having the Kadec-Klee property and \(G\) is a nonempty closed boundedly relatively weakly compact subset of \(X,\) then the set of all \(A\in \mathcal K_ G(X)\) for which the problem \(\min(A,G)\) is well posed contains a dense \(G_ \delta\) subset of \(\mathcal K_ G(X)\) (Theorem 2.1). Under the same hypotheses on the space \(X\), if \(G\in \mathcal B(X)\) is relatively weakly compact, then the set of all \(A\in \mathcal K(X)\) for which the problem \(\max(A,G)\) is well posed contains a dense \(G_ \delta\) subset of \(\mathcal K(X)\) (Theorem 3.1). Similar results for \(X\) uniformly convex or strongly convex (meaning reflexive, strictly convex and Kadec-Klee) were obtained by \textit{F. S. De Blasi, J. Myjak} and \textit{P. L. Papini} [J. Approximation Theory 70, No. 2, 142--155 (1992; Zbl 0764.41032)], respectively by one of the authors, \textit{Chong Li} [J. Approximation Theory 103, No. 1, 1--17 (2000; Zbl 0952.46012)] and \textit{Chong Li} and \textit{Hong-Kun Xu} [Numer. Funct. Anal. Optimization 23, No. 3--4, 323--331 (2002; Zbl 10…
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 49J53
zbMATH DE Number: 2119279
zbMATH Keywords: nearest points
zbMATH Keywords: farthest points
zbMATH Keywords: generic existence results
zbMATH Keywords: Hausdorff metric
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: s10114-003-0287-x
OpenAlex ID: W2049778175
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DOI: 10.1007/S10114-003-0287-X
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 2119279