Gromov-Hausdorff Distance and Borsuk Number

TL;DR

This paper explores the relationship between Gromov-Hausdorff distance and the Borsuk number, providing formulas and corollaries that connect metric space partitioning with topological conjectures.

Contribution

It introduces an exact formula for Gromov-Hausdorff distance under specific conditions and links it to the Borsuk Conjecture through new theoretical results.

Findings

Derived an exact Gromov-Hausdorff distance formula for certain metric spaces

Established connections between Borsuk number and Gromov-Hausdorff distance

Obtained corollaries linking Lusternik-Schnirelmann and Borsuk problems

Abstract

The aim of this paper is to demonstrate relations between Gromov-Hausdorff distance properties 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 assumptions that the diameter and the cardinality of one space is less than the diameter and the Borsuk number of the other one, respectively. Using Bacon equivalence results between Lusternik-Schnirelmann and Borsuk Problems several corollaries are obtained.