On the center of distances of finite ultrametric spaces

TL;DR

This paper investigates the properties of the center of distances in finite ultrametric spaces, establishing bounds on its size and demonstrating the existence of spaces that attain these bounds.

Contribution

It proves an upper bound on the size of the center of distances in finite ultrametric spaces and constructs examples that achieve this bound.

Findings

The size of the center of distances is at most 1 + floor(log2 n) for n-point ultrametric spaces.

There exist ultrametric spaces where the center of distances reaches this maximum size.

The bounds are tight, as shown by explicit constructions.

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. $