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

Abstract

In the present paper the following Generalized Borsuk Problem is studied: Can a given bounded metric space X be partitioned into a given number m (probably an infinite one) of subsets, each of which has a smaller diameter than X? We give a complete answer to this question in terms of the Gromov-Hausdorff distance from X to a simplex of cardinality m and having a diameter less than X. Here a simplex is a metric space, all whose non-zero distances are the same.

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

“…The following theorem is proved in [12]. Theorem 2.12 ( [12]). Let X be an arbitrary bounded metric space, and m a cardinal number with m ≤ #X.…”

“…In paper [12] we considered Generalized Borsuk Problem: Is it possible to partition a given bounded metric space X into a given number of subsets having strictly less diameter than X has. The following theorem is proved in [12]. Theorem 2.12 ( [12]).…”

“…Recently the authors discovered, see [12], that Generalized Borsuk Problem (the same question for an arbitrary bounded metric space X, which is not necessary a subset of R n ) can be solved by means of the Gromov-Hausdorff distance between X and so-called simplexes, i.e., metric spaces all whose nonzero distances are the same (see Theorem 2.12 below).…”

“…Some additional characteristics of the bounded metric spaces were defined, and in this terms either exact formulas for the Gromov-Hausdorff distance to an arbitrary simplex, or an exact upper and low estimates for these distances were obtained. 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].…”

The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces

“…This case turns out to be of special interest due to several reasons. In the case of finite metric space X the distances from X to simplexes permit to reproduce the edges lengths of a minimal spanning tree of X, see [12]; these distances turn out to be useful in investigation of isometries of the Gromov-Hausdorff space, see [9]; in terms of those distances the generalized Borsuk problem can be solved, see [6].…”