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.

Gromov--Hausdorff Distance to Simplexes

“…Notice that ε(X) ≤ diam X, and for a bounded X the equality holds, iff X is a simplex. Corollary 2.5 immediately implies the following result that is proved in [2].…”

“…For calculations of the Gromov-Hausdorff distances one can use the following simple relations, whose proofs can be found in [2].…”

“…In addition, these distances play an important role in investigation of the symmetry group of the space M, see [8]. In [2] we have calculated and estimated the distances between finite simplexes and compact spaces. In particular, it gives us a possibility to construct two non-isometric finite metric spaces with equal distances to all finite simplexes.…”

“…In [2] a similar theory for compact metric spaces is worked out. Here we generalize the results from [2] to any bounded metric space, also, we simplify some proofs from [2].…”

Solution to Generalized Borsuk Problem in Terms of the Gromov-Hausdorff Distances to Simplexes

“…Notice that paper [3] and its generalization [12] are devoted to the calculation of this distance. Relations between the distances of this type and other geometrical problems is also demonstrated in paper [13], where in terms of the distances from a finite metric space X to finite simplexes the edges lengths of a minimal spanning tree constructed on X are calculated.…”