Continuous position and existence sets

Abstract

As ubset M of a linear space is in continuous position pro vided each convex open set meeting M cannot touch M at another position. Every convex set is in continuous position. The present paper is concerned with the re verse 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 Chebyshe vs et must be convex. We p resent tw od istinct attacks of the problem. The first method involves the theory of existence sets and leads (a.o.) to an af firmati ve solution of our problem in superrefle xive Banach spaces. The second method involves some purely topological (non-metric) tech- niques and leads a.o. to an af firmative s olution of the problem for closed and boundedly weakly compact sets in arbitrary Banach spaces and for locally compact sets in locally con- vex spaces.