2017/09/24 by Alexey Pushnyakov, Pushnyakov, Alexey
Mathematics · #Advanced Topology and Set Theory #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1709.08280
openalex publication_date 2017/09/24 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28
A compact metric space (X, \ρ) is given. Let \μ be a Borel measure on\nX. By r-cluster we mean a measurable subset of X with diameter at most\nr. A family of k 2r-clusters is called a r-cluster structure of order\nk if any two clusters from the family are separated by a distance at least\nr. By measure of a cluster structure we mean a sum of clusters measures from\nthe cluster structure. Using the Blaschke selection theorem one can prove that\nthere exists a cluster structure \X^* of maximum measure. We study\ndependence \μ(\X^*) on distance distribution. The main issue is to\nfind restrictions for distance distribution which guarantee that\n\μ(\X^*) is close to \μ(X). We propose a discretization of\ndistance distribution and in terms of this discretization obtain a lower bound\nfor \μ(\X^*).\n