Center of distances of ultrametric spaces generated by labeled trees

TL;DR

This paper investigates the properties of the center of distances in ultrametric spaces generated by labeled trees, providing conditions for when the center includes the diameter of the space.

Contribution

It characterizes the center of distances in ultrametric spaces from labeled trees and establishes necessary and sufficient conditions for including the diameter.

Findings

C(X) is either {0} or {0, diam X} for these spaces.

Conditions for diam X to be in C(X) are identified.

Provides a complete description of the center of distances in this context.

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 that the equalities $C(X)=\{0\}$ or $C(X)=\{0,\operatorname{diam}X\} $ hold if $(X,d)$ is an ultrametric space generated by labeled trees. The necessary and sufficient conditions under which $\operatorname{diam} X\in C(X)$ are found.