On the center of distances of finite ultrametric spaces

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 $n\ge 1$ there exists a finite ultrametric space $(Y,\rho )$ such that $|Y|=n$ and $|C(Y)|=1+{\log }_2\lfloor n\rfloor .$