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On the center of distances of finite ultrametric spaces

Oleksiy Dovgoshey, Olga Rovenska

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

On the center of distances of finite ultrametric spaces

Abstract

The center of distances of a metric space is the set of all for which the equation has a solution for each . We prove the inequality for all finite ultrametric spaces which have exactly points. It is also shown that for every integer there exists a finite ultrametric space such that and

Paper Structure

This paper contains 4 sections, 16 theorems, 170 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The upper bound of number of points of $C(X)$
  4. Concluding Remarks and Expected Results

Key Result

Proposition 2.3

Let $(X,d)$ be an ultrametric space. Then the equality holds for each open ball $B_r(c) \subseteq X$ and every point $a \in B_r(c)$.

Figures (1)

  • Figure 1: The four-point ultrametric spaces $(X_4,d_4)$ and $(Y_4,\rho_4)$ and their representing trees.

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 37 more