Gromov--Hausdorff Distance to Simplexes

Abstract

Geometric characteristics of metric spaces that appear in formulas of Gromov-Hausdorff distances from these spaces to so-called simplexes, i.e., to the metric spaces, all whose non-zero distances are the same are studied. The corresponding calculations essentially use geometry of partitions of these spaces. In the finite case, it gives the lengths of minimal spanning trees [1]. 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

“…We need the following two Theorems from [12], which the main results of the present paper is based on. Theorem 1.2 ([12, Theorem 2.1]).…”

“…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.…”

The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces

“…We use short notations for α + m (X) and d − m (X), because in the formulas given below these two values appears more often than there "twins" α − m (X) and d + m (X). Theorem 1.6 ( [5]). Let X be an arbitrary bounded metric space, and m = #λ∆ ≤ #X.…”

“…In paper [7] the distances between simplexes and compact metric spaces are calculated in several particular cases. Later on these results are generalized to the case of arbitrary bounded metric spaces, see [5]. Namely, several additional characteristics of the bounded metric spaces were introduced, and in terms of those characteristic either exact formulas for the Gromov-Hausdorff distance to simplexes were written, or exact lower and upper estimates for these distances were given.…”

“…In the present paper the formulas from [5] for the Gromov-Hausdorff distance from an arbitrary bounded metric space X to a simplex get a geometrical interpretation, that permits to rewrite them in a more convenient from in the case of simplexes having at most the same cardinality as the space X, see Theorem 2.5. As an application, an exact formulas for the distances from a simplex to an arbitrary finite ultrametric space are obtained, see Theorem 3.3.…”