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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.14317 |
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| _version_ | 1866918298138968064 |
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| author | Martini, Horst Martín, Pedro Spirova, Margarita |
| author_facet | Martini, Horst Martín, Pedro Spirova, Margarita |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14317 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Chebyshev sets and ball operators Martini, Horst Martín, Pedro Spirova, Margarita Metric Geometry 41A50, 41A61, 46B20, 52A21 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. |
| title | Chebyshev sets and ball operators |
| topic | Metric Geometry 41A50, 41A61, 46B20, 52A21 |
| url | https://arxiv.org/abs/2601.14317 |