On the relationships of cluster measures and distributions of distances in compact metric spaces

A compact metric space with a bounded Borel measure is considered. Any measurable set of diameter not exceeding $r$ is called $r$-cluster. The existence of a collection consisting of a fixed number of $2r$-clusters possessing the following properties is investigated: the clusters are located at the distance $r$ from each other and the collection measure (the total measure of the clusters in the collection) is close to the measure of the entire space. It is proved that there exists a collection with a maximum measure among such collections. The concept of $r$-parametric discretization of the distribution of distances into short, medium, and long distances is defined. In terms of this discretization, a lower bound on the measure of the maximum-measure collection is obtained.