Gromov--Hausdorff Distance Between Euclidean Unit Balls

Abstract

What is the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions, denoted by $d_\gh(B^m,B^n)$, for $m>n$? Note that the lower bound coming from the stability of persistent homology is zero, since all balls possess identical (trivial) persistent homology. To establish non-trivial lower bounds, we exploit the Borsuk--Ulam theorem. For any $m>n \ge 1$, we prove that $d_\gh(B^m,B^n)\ge \tfrac{1}{1+\alpha_n}>\frac{1}{2}$, where we determine $\alpha_n$ explicitly. We also prove that $d_\gh(B^m,B^n)\to 1$ as $m\to \infty$ and that $d_\gh(B^m,B^n)<1$ for all finite $m>n\geq 1$.