Calculating Gromov-Hausdorff Distance by means of Borsuk Number

TL;DR

This paper explores the relationship between Gromov-Hausdorff distance and Borsuk number, providing an exact formula under certain conditions and deriving corollaries through topological problem equivalences.

Contribution

It introduces a novel connection between Gromov-Hausdorff distance and Borsuk number, deriving an exact formula under specific assumptions.

Findings

Exact formula for Gromov-Hausdorff distance under certain conditions

Connection established between Gromov-Hausdorff distance and Borsuk number

Derived corollaries using topological problem equivalences

Abstract

The purpose of this article is to demonstrate the connection between the properties of the Gromov--Hausdorff distance and the Borsuk conjecture. The Borsuk number of a given bounded metric space $X$ is the infimum of cardinal numbers $n$ such that $X$ can be partitioned into $n$ smaller parts (in the sense of diameter). An exact formula for the Gromov--Hausdorff distance between bounded metric spaces is obtained under the assumption that the diameter and cardinality of one space are less than the diameter and Borsuk number of another, respectively. Using the results of Bacon's equivalence between the Lusternik--Schnirelmann and Borsuk problems, several corollaries are obtained.