Calculation of the Gromov–Hausdorff distance using the Borsuk number

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 derived 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 P. Bacon equivalence results between Lusternik–Schnirelmann and Borsuk problems, several corollaries are obtained.