Projections onto closed convex sets in Hilbert spaces
type: Item
instance of: scholarly article
title: Projections onto closed convex sets in Hilbert spaces
zbMATH Open document ID: 1389.52007
published in: Acta Mathematica Hungarica
publication date: 2017-11-10T00:00:00Z
review text: Let \(X\) be a real Hilbert space. A set \(C\) in \(X\) is a Chebyshev set if for each point \(x\) in \(X\), there is a unique point of \(C\) that is nearest to \(x\); that is, there is a point \(q\) in \(C\) such that \(\| x- q\| <\| x-y\| \) for all \(y \in {C \setminus \{q\}}\). A mapping \(N:X\rightarrow C\) which takes each point \(x\in X\) to its nearest point \(q\) in \(C\) is called the metric projection. In this paper, the authors give necessary and sufficient algebraic conditions for a mapping \(F : X \rightarrow X\) with a closed image set to be the metric projection mapping onto a closed convex set. They provide examples that illustrate the necessity of each of the conditions. The characterizations given in the paper generalize several known results related to projections onto closed convex sets.
Mathematics Subject Classification ID: 52A27
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 6806130
zbMATH Keywords: Hilbert space
zbMATH Keywords: closed convex set
zbMATH Keywords: metric projection
author: András Domokos
author: John M. Ingram
author: M. M. Marsh
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: s10474-017-0691-9
OpenAlex ID: W2591476178
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DOI: 10.1007/S10474-017-0691-9
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description: scientific article; zbMATH DE number 6806130