Chebyshev sets and ball operators

TL;DR

This paper explores properties of Chebyshev sets in normed spaces using ball operators, revealing new relationships, conditions for singleton Chebyshev sets, and geometric descriptions relevant for algorithms.

Contribution

It introduces new properties of Chebyshev sets via ball operators, characterizes when Chebyshev sets are singletons, and describes the ball hull of finite planar sets.

Findings

Chebyshev set of a bounded set contains that of some completion

Necessary and sufficient conditions for Chebyshev set to be a singleton

Complete geometric description of the ball hull of finite planar sets

Abstract

The Chebyshev set of a bounded set $K$ in a normed space is the set of centers of all minimal enclosing balls of $K$. We use the concept of ball intersection and ball hull operators to derive new properties of Chebyshev sets in normed spaces. These results give a better picture on how Chebyshev sets, ball intersections, ball hulls, and completions of bounded sets are related to each other. It is shown that the Chebyshev set of a bounded set $K$ always contains the Chebyshev set of some completion of $K$. Moreover, for a special class of sets we obtain a necessary and sufficient condition that the Chebyshev set of the respective set is a singleton. We obtain new results on critical sets of Chebyshev centers, and for that purpose, surprisingly, notions from the combinatorial geometry of convex bodies play an essential role. Also we give a complete geometric description of the ball hull of a finite planar set. This can be taken as starting point for algorithmical constructions of the ball hull of such sets.