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

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