Cosuns in \(\ell ^ p(n)\), \(1\leq p<\infty \).
type: Item
instance of: scholarly article
title: Cosuns in \(\ell ^ p(n)\), \(1\leq p<\infty \).
zbMATH Open document ID: 0658.41012
DOI: 10.1016/0021-9045(88)90005-6
author: U. Westphal
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: For a nonempty subset K of a normed space X and \(x\in X\) put \(R_ K(x)=\{k\in K:\| k-k'\| \leq \| x-k'\|\), \(\forall k'\in K\}\) and \(A_ K(x)=\{z\in X:\| z-k\| \leq \| z+\lambda (x-z)-k\|\), \(\forall k\in K\), \(\forall \lambda >0\}\). The elements of \(R_ K(x)\) (if exist) are called elements of best coapproximation of x in K. The set K is called coproximinal if \(R_ K(x)\neq \emptyset\), \(\forall x\in X\), and a cosun if \(A_ K(x)\cap K\neq \emptyset,\) \(\forall x\in X\). Every cosun is a coproximinal set and if X is finite dimensional, every coproximinal set is a cosun. The coproximinal sets are also related to the optimal sets in the sense of \textit{B. Beauzamy} and \textit{B. Maurey} [J. Funct. Anal. 24, 107-139 (1977; Zbl 0344.46049)]. The paper is devoted mainly to characterizations of cosuns in \(\ell^ p(n)\), \(l\leq p<\infty\), \(p\neq 2\), \(n\geq 3\). These can be characterized as the norm-convex (in the sense of Menger) subsets of \(\ell^ 2(n)\)-normals having at most two nonzero coordinates. The results obtained by the author complete the results of \textit{L. Hetzelt} [Dissertation, Univ. of Erlangen-Nürnberg (1981) and Acta Math. Hung 45, 53-68 (1985; Zbl 0592.41043)].
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 4075727
zbMATH Keywords: cosun
zbMATH Keywords: coproximinal set
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90005-6
OpenAlex ID: W2070047865
cites work: Contractive projections in \(L_ p\)-spaces
cites work: Q4153188
cites work: Points minimaux et ensembles optimaux dans les espaces de Banach
cites work: Über die beste Approximation im \({\mathbb{R}}^ n\)
cites work: Q3943398
cites work: Q5610277
cites work: Subspaces of l p,n Spaces
cites work: COMBINATORIAL GEOMETRY AND CONVEXITY CLASSES
cites work: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces
cites work: Nonexpansive projections on subsets of Banach spaces
cites work: The extension of contractions and the intersection of balls in Banach spaces
cites work: Fixed points of quasi-nonexpansive mappings
cites work: Contractive projections on an \(L_ 1\) space
cites work: Q5657185
cites work: Fixpunktmengen von Kontraktionen in endlichdimensionalen normierten Räumen
cites work: On suns and cosuns in finite dimensional normed real vector spaces
cites work: Approximation and strong approximation in normed spaces via tangent functionals
cites work: Best coapproximation in normed linear spaces
cites work: APPROXIMATIVE PROPERTIES OF SETS IN NORMED LINEAR SPACES
cites work: Q3937146
description: scientific article; zbMATH DE number 4075727