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arXiv · 2603.25850

On the center of distances of finite ultrametric spaces

Oleksiy Dovgoshey·Olga Rovenska
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Abstract

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 $(Y,\rho)$ such that $|Y| = n$ and $|C(Y)| = 1 + \log_2 \lfloor n \rfloor. $

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Oleksiy Dovgoshey, Olga Rovenska. 2026-03-26. On the center of distances of finite ultrametric spaces. https://arxiv.org/abs/2603.25850

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