Antiproximal sets
type: Item
instance of: scholarly article
title: Antiproximal sets
zbMATH Open document ID: 0618.41028
DOI: 10.1016/0021-9045(87)90102-X
author: Michael Edelstein
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: A set C in a Banach space X is said to be antiproximal if no point in \(X\setminus C\) has a nearest point in S. If C is a closed bounded symmetric convex body in X, then, it is readily seen, C is antiproximal if and only if the closed unit ball of X is antiproximal in the norm induced by C. Denote by U (respectively D) the unit ball of \(c_ 0\) equipped with the usual norm (respectively Day's norm). \textit{S. Cobzaş} [Math. Rev. Anal. Numér. Téor. Approximation, Math. 2, 137-141 (1973)] has proved that D is antiproximal in \(c_ 0\) with respect to the usual norm. In the present paper the author constructs an isomorphism \(A: c_ 0\to c_ 0\), \(A\neq id.\), such that A(U) is antiproximal in \(c_ 0\) with respect to Day's norm.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 4002741
zbMATH Keywords: proximinal sets
zbMATH Keywords: equivalent renorming
zbMATH Keywords: antiproximal
reviewed by: Ioan Şerb
MaRDI profile type: Publication
cites work: Some results on nearest points and support properties of convex sets in \(c_ 0\)
cites work: Local Uniform Convexity of Day's Norm on c 0 (Γ)
description: scientific article; zbMATH DE number 4002741