Properties of sets that have a continuous selection of the operator \(P^{\delta}\)
type: Item
instance of: scholarly article
title: Properties of sets that have a continuous selection of the operator \(P^{\delta}\)
zbMATH Open document ID: 0729.46011
DOI: 10.1007/BF01139608
author: Igor G. Tsar'kov
published in: Mathematical Notes
publication date: 1990-01-01T00:00:00Z
review text: If X is a Banach space, \(M\subset X\), \(x\in X\) and \(\delta >0\), then \(P_ M^{\delta}(x)=\{y\in M;\quad \| x-y\| \leq dist(x,M)+\delta \}.\) The interplay of the following properties a), b), c) is investigated: a) the existence of continuous selection for the operator \(x\to P_ M^{\delta}(x),\) b) approximate compactness of M, c) convexity properties of M. Typical result (Corollary 2). If \(X^*\) is strictly convex, \(M\subset X\) is approximately compact and \(P_ M^{\delta}\) admits a continuous selection for every \(\delta >0\), then M is convex.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 4205137
zbMATH Keywords: approximate compactness
zbMATH Keywords: continuous selection
reviewed by: V. Zizler
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01139608
OpenAlex ID: W2059146472
description: scientific article; zbMATH DE number 4205137