Interdependence of clusters measures and distance distribution in compact metric spaces

TL;DR

This paper investigates how the measure of optimal cluster structures in compact metric spaces depends on the distribution of distances, providing bounds based on discretized distance data.

Contribution

It introduces a discretization approach to analyze the relationship between distance distribution and maximum cluster measure in compact metric spaces.

Findings

Existence of a maximum measure cluster structure proven via Blaschke selection theorem.

Derived lower bounds for maximum cluster measure based on discretized distance distribution.

Establishes conditions under which the maximum cluster measure approaches the total measure of the space.

Abstract

A compact metric space $(X, \rho)$ is given. Let $\mu$ be a Borel measure on $X$. By $r$-cluster we mean a measurable subset of $X$ with diameter at most $r$. A family of $k$ $2r$-clusters is called a $r$-cluster structure of order $k$ if any two clusters from the family are separated by a distance at least $r$. By measure of a cluster structure we mean a sum of clusters measures from the cluster structure. Using the Blaschke selection theorem one can prove that there exists a cluster structure $\mathcal{X}^*$ of maximum measure. We study dependence $\mu(\mathcal{X}^*)$ on distance distribution. The main issue is to find restrictions for distance distribution which guarantee that $\mu(\mathcal{X}^*)$ is close to $\mu(X)$. We propose a discretization of distance distribution and in terms of this discretization obtain a lower bound for $\mu(\mathcal{X}^*)$.