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arXiv · 2601.14317

Chebyshev sets and ball operators

Horst Martini·Pedro Martín·Margarita Spirova
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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.

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Horst Martini, Pedro Martín, Margarita Spirova. 2026-01-19. Chebyshev sets and ball operators. https://arxiv.org/abs/2601.14317

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