Smooth antiproximinal sets
type: Item
instance of: scholarly article
title: Smooth antiproximinal sets
zbMATH Open document ID: 0921.46013
DOI: 10.1007/BF02317791
author: V. S. Balaganskij
published in: Mathematical Notes
publication date: 1999-09-17T00:00:00Z
review text: Let \(S(X)\) be the unit sphere of the real Banach space \(X\). A nonempty closed convex set \(M\subsetneqq X\) is said to be smooth if the conditions \(x\in M\), \(f,g\in S(X^*)\), \(f(x)=\sup\{ f(y): y\in M\}\), \(g(x)=\sup\{f(y):y\in M\}\) imply that \(f=g\). The set \(M \subsetneqq X\) is said to be antiproximinal if no point \(x\in X\setminus A\) has a nearest point in \(M\). At the beginning the author shows that a Banach space contains a convex smooth compact set if and only if \(X\) is separable. Using this fact one obtains the main result: Let \(X\) be a separable Banach space. If \(X\) contains a nonempty convex closed bounded antiproximinal set then \(X\) necessarily contains a nonempty smooth convex closed bounded antiproximinal set. In particular \(C[0,1]\), \(c_0\), \(c\) contain a smooth bounded convex centrally symmetric antiproximinal body.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 1262573
zbMATH Keywords: antiproximinal sets
zbMATH Keywords: smooth sets
zbMATH Keywords: strictly convex sets in Banach spaces
zbMATH Keywords: centrally symmetric antiproximinal body
reviewed by: Ioan Şerb
MaRDI profile type: Publication
cites work: Geometry of Banach spaces. Selected topics
cites work: Q5332803
description: scientific article; zbMATH DE number 1262573