On the center of distances of finite ultrametric spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911547345862656 |
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| author | Dovgoshey, Oleksiy Rovenska, Olga |
| author_facet | Dovgoshey, Oleksiy Rovenska, Olga |
| contents | The center of distances of a metric space $(X,d)$ is the set $C(X)$ of all $t\in \mathbb R^+$ for which the equation $d(x,p)=t$ has a solution for each $p\in X$. We prove the inequality $|C(X)| \le 1 + \lfloor \log_2 n \rfloor$ for all finite ultrametric spaces $(X,d)$ which have exactly $n$ points. It is also shown that for every integer $n \geq 1$ there exists a finite ultrametric space $(Y,ρ)$ such that $|Y| = n$ and $|C(Y)| = 1 + \log_2 \lfloor n \rfloor. $ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25850 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the center of distances of finite ultrametric spaces Dovgoshey, Oleksiy Rovenska, Olga Metric Geometry General Topology 54E35 The center of distances of a metric space $(X,d)$ is the set $C(X)$ of all $t\in \mathbb R^+$ for which the equation $d(x,p)=t$ has a solution for each $p\in X$. We prove the inequality $|C(X)| \le 1 + \lfloor \log_2 n \rfloor$ for all finite ultrametric spaces $(X,d)$ which have exactly $n$ points. It is also shown that for every integer $n \geq 1$ there exists a finite ultrametric space $(Y,ρ)$ such that $|Y| = n$ and $|C(Y)| = 1 + \log_2 \lfloor n \rfloor. $ |
| title | On the center of distances of finite ultrametric spaces |
| topic | Metric Geometry General Topology 54E35 |
| url | https://arxiv.org/abs/2603.25850 |