Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space
type: Item
instance of: scholarly article
title: Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space
zbMATH Open document ID: 0818.54026
DOI: 10.1007/BF01292766
published in: Monatshefte für Mathematik
publication date: 1995-04-03T00:00:00Z
full work available at URL: https://eudml.org/doc/178671
review text: The paper contains two main theorems and several auxiliary results. Let \(E\) denote a real Banach space with norm \(\|\cdot \|\) and \(S(E)\) the family of all nonempty, compact, starshaped subsets of \(E\) which is a complete metric space when endowed with the Hausdorff metric. Let \(\pi_ X\) be the metric projection onto \(X\), i.e. the mapping which associates to each \(a\in E\) the set of all points in \(X\) closest to \(a\) and let \(A(X)\) denote the ambiguous locus of \(X\), i.e. \(A(X)= \{a\in E\): \(\text{diam } \pi_ X (a)> 0\}\). In a complete metric space \(M\) the complement of any set of the first Baire category is called a residual subset of \(M\) and its elements are called typical elements of \(M\). A subset \(X\) of a metric space \(M\) is everywhere uncountable if for every \(a\in M\) and \(r>0\) the set \(X\cap B_ M (a,r)\) is nonempty and uncountable. A set \(X\) is starshaped with respect to a point \(u\in X\) if the closed segment \([u,x]\) is contained in \(X\) for every \(x\in X\) and \(\ker X\) is the set of all points with respect to which \(X\) is starshaped. The authors prove that a set \(A\) which denotes the set of all \(X\in S(E)\) with ambiguous locus \(A(X)\) everywhere uncountable in a strictly convex separable Banach space \(E\) with \(\dim E\geq 2\), is a residual subset of \(S(E)\). They also prove that a typical element of \(S(E)\) has a kernel consisting of a single point and a set of directions dense in the unit sphere of \(E\).
reviewed by: Semin Akdoǧan
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 733128
zbMATH Keywords: starshaped subsets
zbMATH Keywords: complete metric space
zbMATH Keywords: metric projection
zbMATH Keywords: ambiguous locus
zbMATH Keywords: first Baire category
zbMATH Keywords: residual subset
zbMATH Keywords: typical elements
zbMATH Keywords: kernel
author: F. S. de Blasi
author: Petar S. Kenderov
author: Józef Myjak
MaRDI profile type: Publication
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OpenAlex ID: W1969709641
description: scientific article; zbMATH DE number 733128