Chebyshev sets and ball operators
Horst Martini, Pedro Martín, Margarita Spirova
TL;DR
The paper examines how Chebyshev sets interact with ball hulls, ball intersections, and set completions in finite-dimensional normed spaces. By developing and exploiting the ball hull and ball intersection operators, it proves containment relations between $Ch(K)$ and $Ch(K^c)$ for completions, characterizes when $Ch(K)$ is a singleton in the planar centrable case via critical points and meanstream centers, and provides a complete geometric description of the ball hull for finite planar sets as $Ch(K)=\operatorname{bh}(K)$ described by minimal circular arcs. It introduces the critical set and meanstream Chebyshev centers to connect Chebyshev centers with inner illuminating systems from convex geometry, yielding both structural insights and a practical planar algorithmic starting point. The work thus unifies circumballs, ball hulls, and completions, clarifying when Chebyshev centers lie inside convex hulls and offering explicit planar constructions for ball hulls.
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.