The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces

Abstract

In the present paper we investigate the Gromov-Hausdorff distances between a bounded metric space X and so called simplex, i.e., a metric space all whose non-zero distances are the same. In the case when the simplex's cardinality does not exceed the cardinality of X, a new formula for this distance is obtained. The latter permits to derive an exact formula for the distance between a simplex and an ultrametric space.

The Gromov--Hausdorff Distance between Simplexes and Two-Distance Spaces

“…Just those formulas permitted to discover a relations between the distances to simplexes and Borsuk problem, see [12]. After that, using a geometrical interpretation, formulas from [18] have been rewritten in a more convenient way that gives an opportunity to calculate the distances between simplexes and an arbitrary finite ultrametric space [19].…”

“…In the present paper, formulas from [19] are used to calculate the Gromov-Hausdorff distances from simplexes to finite 2-distance spaces (Theorem 2.11). That permits to obtain a complete solution to generalized Borsuk problem for such spaces.…”