On a generalized best approximation problem
type: Item
instance of: scholarly article
title: On a generalized best approximation problem
zbMATH Open document ID: 0912.41015
published in: Journal of Approximation Theory
publication date: 1999-05-16T00:00:00Z
full work available at URL: https://polipapers.upv.es/index.php/AGT/article/view/1994
review text: Let \(C\) be a closed bounded convex subset of a Banach space \(E\) which has the origin of \(E\) as an interior point and let \(P_C\) denote the Minkowski functional with respect to \(C.\) For a nonempty closed set \(X\subset E\) and a point \(u\in E\) a minimization problem which consists in proving the existence of a point \(\widetilde x\in X\) such that \(P_C({\widetilde x}-u)= \inf\{P_C(x-u)\mid x\in X\}\) is studied . If \(\widetilde x\) is unique and every minimizing sequence converges to \(\widetilde x\), the minimization problem for \(u\in E, X\subset E\) is called well posed. It is proved that if \(E_0(X)\) is the set of all \(u\in E\) for which the minimization problem is well posed and the modulus of convexity with respect to \(P_C\) is strictly positive, then \(E_0(X)\) is a dense \(G_\delta\) subset of \(E\). It should be noted that for most bounded subsets \(X\subset E,\) the set \(E\backslash E_0(X)\) is dense in \(E.\)
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 1214032
zbMATH Keywords: best approximation
zbMATH Keywords: Banach space
zbMATH Keywords: Minkowski functional
zbMATH Keywords: modulus of convexity
zbMATH Keywords: minimization problem
author: F. S. de Blasi
author: Józef Myjak
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
OpenAlex ID: W2066054759
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DOI: 10.1006/JATH.1998.3177
description: scientific article; zbMATH DE number 1214032