Continuous position and existence sets
type: Item
instance of: scholarly article
title: Continuous position and existence sets
zbMATH Open document ID: 0635.46011
DOI: 10.1007/BF01462888
author: Marcel van de Vel
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164486
review text: A subset M of a linear space is in continuous position provided each convex open set meeting M cannot ``touch'' it at some other place. Every convex set is in continuous position. The present paper is concerned with the reverse problem whether a closed set in continuous position must be convex (closedness is essential). This problem is similar to - and connected with - the problem whether every Chebyshev set must be convex. We present two distinct ways of attacking the problem. The first method involves the theory of existence sets and leads a.o. to an affirmative solution of our problem in superreflexive Banach spaces. The second method involves some purely topological ideas and leads a.o. to an affirmative solution of the problem for closed and boundedly weakly compact sets in arbitrary Banach spaces and for locally compact sets in locally convex spaces.
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 4034387
zbMATH Keywords: continuous position
zbMATH Keywords: whether a closed set in continuous position must be convex
zbMATH Keywords: whether every Chebyshev set must be convex
zbMATH Keywords: existence sets
zbMATH Keywords: superreflexive Banach spaces
zbMATH Keywords: boundedly weakly compact sets in arbitrary Banach spaces
zbMATH Keywords: locally compact sets in locally convex spaces
zbMATH Keywords: nearest point
zbMATH Keywords: smooth norm
zbMATH Keywords: uniformly rotund norm
MaRDI profile type: Publication
cites work: Geometry of Banach spaces. Selected topics
cites work: Some results on nearest points and support properties of convex sets in \(c_ 0\)
cites work: Q4080169
cites work: Convexity of Chebyshev sets
cites work: Q3310512
cites work: APPROXIMATIVE PROPERTIES OF SETS IN NORMED LINEAR SPACES
OpenAlex ID: W2151605493
description: scientific article; zbMATH DE number 4034387