Center of distances of ultrametric spaces generated by labeled trees

TL;DR

We study the center of distances $C(X)$ for ultrametric spaces generated by labeled trees ($${\bf UT}$$). The authors prove a dichotomy: for any ${\bf UT}$-space $(X,d)$ with $|X|\ge 2$, $C(X)={\{0\}}$ or $C(X)={\{0,\operatorname{diam} X\}}$, and the latter occurs precisely when $\operatorname{diam} X\in D(X)$. They further connect this to centered spheres and the geometry of the diametrical graph $G_X$, showing that $C(X)={0,\operatorname{diam} X}$ is equivalent to $G_X$ having a spanning star and to the possibility of $X$ being a centered sphere. The results relate the center of distances to weak similarities and to graph-theoretic structures, offering a structural understanding of UT-spaces and implications for p-adic-like ultrametrics. This framework advances the geometric characterization of centers of distances in ultrametric spaces generated by labeled trees, with potential implications for applications in areas using ultrametric and p-adic models.

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)=\{\operatorname{diam} X,0\} $ 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.

Figures (3)

• Figure 1: The space $(X,d)$ is isometric to the ultrametric space $(V(P),d_l)$ generated by labeled path $P=P(l)$.
• Figure 2: A spanning star graph $St_X$ of the diametrical graph $G_X$ of the space $(X,d)$ depicted by Figure \ref{['fig1']}.
• Figure 3: A non-equidistant three-point ultrametric space $(X_3,d)$.

Theorems & Definitions (59)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Example 1: The $p$-adic ultrametric
• Proposition 1
• Definition 4
• Proposition 2
• Definition 5
• Definition 6