Continuous position and existence sets
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.
whether a closed set in continuous position must be convex
whether every Chebyshev set must be convex
boundedly weakly compact sets in arbitrary Banach spaces
locally compact sets in locally convex spaces
scientific article; zbMATH DE number 4034387