Projections onto convex cones in Hilbert space
type: Item
instance of: scholarly article
title: Projections onto convex cones in Hilbert space
zbMATH Open document ID: 0728.41034
DOI: 10.1016/0021-9045(91)90067-K
author: John M. Ingram
author: M. M. Marsh
published in: Journal of Approximation Theory
publication date: 1991-01-01T00:00:00Z
review text: A subset C of a real Hilbert space H is called a Chebyshev set if each \(x\in H\) has a unique point p(x) of best approximation in C. It is not difficult to see that if C is a closed convex cone, then p is (1) idempotent, (2) non-negative homogeneous, and (3) \(p(x+y)=p(x)+p(y)\) if and only if \(<p(x),y>=<p(x),p(y)>=<x,p(y)>.\) The authors investigate the converse question, namely whether conditions (1), (2), and (3) on a mapping n: \(H\to H\) automatically force \(K=n(H)\) to be a closed convex cone with projection n. Amont other things, they show that \(K=n(H)\) is a closed, convex cone and n is the projection mapping of H onto K if and only if the function n is continuous and satisfies the three conditions. The paper concludes with a discussion of the open question of whether the assumption of continuity in the above result can be omitted.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 4202997
zbMATH Keywords: Chebyshev set
reviewed by: Richard M. Aron
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4202997