Gromov--Hausdorff Distance Between Euclidean Unit Balls

Abstract: What is the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions, denoted by $d_\gh(B<sup>m,B<sup>n)$, for $m&gt;n$? 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 $d_\gh(B^{m,B^n)\ge} \frac{\sqrt{n+1}}{\sqrt{n+1}+\sqrt{n}} &gt; \frac{1}{2}$ for $m&gt;n$, and that $d_\gh(B^{m,B^n)\to} 1$ as $m\to \infty$. Finally, we prove that $d_\gh(B<sup>m,B<sup>n)&lt;1$ for all finite $m&gt;n\geq 1$.