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.

References in corpus (1)

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

Cited by in corpus (2)

• The Gromov--Hausdorff Distance between Simplexes and Two-Distance Spaces
• The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces