Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes

Abstract: In the present paper we investigate geometric characteristics of compact metric spaces, which can be described in terms of Gromov-Hausdorff distances to simplexes, i.e., to finite metric spaces such that all their nonzero distances are equal to each other. It turns out that these Gromov-Hausdorff distances depend on some geometrical characteristics of finite partitions of the compact metric spaces; some of the characteristics can be considered as a natural analogue of the lengths of edges of minimum spanning trees. As a consequence, we constructed an unexpected example of a continuum family of pairwise non-isometric finite metric spaces with the same distances to all simplexes.