Interdependence of clusters measures and distance distribution in compact metric spaces

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 $\mu(\mathcal{X}^*)$.