Convexities and approximative compactness and continuity of metric projection in Banach spaces
type: Item
instance of: scholarly article
title: Convexities and approximative compactness and continuity of metric projection in Banach spaces
zbMATH Open document ID: 1190.46018
published in: Journal of Approximation Theory
publication date: 2010-01-08T00:00:00Z
review text: The paper is concerned with the relations between various geometric properties of a Banach space \(X\) -- strong convexity, nearly strong convexity, the property of being very convex or nearly very convex -- and the continuity properties of the metric projection, as upper semi-continuity (usc), strong Wijsman-Zhang usc, on closed convex subsets of \(X.\) The relevance of the approximative compactness for the continuity of the metric projection is also emphasized. For instance, in a nearly strongly convex Banach space, a closed convex subset \(A\) of \(X\) is proximinal with usc metric projection \(P_A\) iff \(A\) is approximatively compact. The paper also contains a representation of the metric projection on \(w^*\)-closed hyperplanes in \(X^*\) and conditions ensuring its upper semi-continuity. The obtained results improve some previous results obtained by several authors.
reviewed by: Costică Mustăţa
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 5655048
zbMATH Keywords: metric projection
zbMATH Keywords: set-valued maps
zbMATH Keywords: upper semi-continuity
zbMATH Keywords: geometry of Banach spaces
zbMATH Keywords: Chebyshev sets
zbMATH Keywords: proximinal sets
zbMATH Keywords: approximative compactness
author: Zihou Zhang
author: Zhongrui Shi
MaRDI profile type: Publication
full work available at URL: j.jat.2009.01.003
OpenAlex ID: W2024737131
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DOI: 10.1016/J.JAT.2009.01.003
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description: scientific article; zbMATH DE number 5655048