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.1709.08286
openalex publication_date 2017/09/24 · openalex created_date 2022/10/02 · 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. In our previous work we showed that under some\nparametric restrictions for distance distribution measure of maximal cluster\nstructure \μ(\X)^* is close \μ(X) and lower bound for\n\μ(\X)^* converges to \μ(X) when corresponding parameters tend\nto 0. However, this bound asymptotically unimprovable. We propose an additional\nrestriction for distance distribution that is responsible for balance of\ncluster's measure in cluster structure. This restriction allows to\nsignificantly improve previous bound in asymptotic sense.\n