Gromov--Hausdorff Distance Between Euclidean Unit Balls

What is the Gromov–Hausdorff distance between Euclidean unit balls of different dimensions, denoted by $d_{\text{gh}}(B^m,B^n)$, 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 $n\ge 1$, we prove that $d_{\text{gh}}(B^m,B^n)\ge \frac{\sqrt{n+1}}{\sqrt{n+1}+\sqrt{n}}>\frac{1}{2}$ for $m>n$, and that $d_{\text{gh}}(B^m,B^n)\to 1$ as $m\to \mathrm{\infty }$. Finally, we prove that $d_{\text{gh}}(B^m,B^n)<1$ for all finite $m>n\ge 1$.