Interdependence of clusters measures and distance distribution in compact metric spaces
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A compact metric space (X, ρ) is given. Let μ 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 X^* of maximum measure. We study dependence μ(X^*) on distance distribution. The main issue is to find restrictions for distance distribution which guarantee that μ(X^*) is close to μ(X). We propose a discretization of distance distribution and in terms of this discretization obtain a lower bound for μ(X^*).
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