On improved bound for measure of cluster structure in compact metric spaces

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. In our previous work we showed that under some parametric restrictions for distance distribution measure of maximal cluster structure $\mu(\mathcal{X})^*$ is close $\mu(X)$ and lower bound for $\mu(\mathcal{X})^*$ converges to $\mu(X)$ when corresponding parameters tend to 0. However, this bound asymptotically unimprovable. We propose an additional restriction for distance distribution that is responsible for balance of cluster's measure in cluster structure. This restriction allows to significantly improve previous bound in asymptotic sense.