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

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