The Gromov-Hausdorff distance between spheres

Abstract

We provide general upper and lower bounds for the Gromov-Hausdorff distance $d_{\mathrm{GH}}({\mathbb{S}}^m,{\mathbb{S}}^n)$ between spheres ${\mathbb{S}}^m$ and ${\mathbb{S}}^n$ (endowed with the round metric) for $0\le m<n\le \infty$. Some of these lower bounds are based on certain topological ideas related to the Borsuk-Ulam theorem. Via explicit constructions of (optimal) correspondences we prove that our lower bounds are tight in the cases of $d_{\mathrm{GH}}({\mathbb{S}}^0,{\mathbb{S}}^n)$, $d_{\mathrm{GH}}({\mathbb{S}}^m,{\mathbb{S}}^{\infty })$, $d_{\mathrm{GH}}({\mathbb{S}}^1,{\mathbb{S}}^2)$, $d_{\mathrm{GH}}({\mathbb{S}}^1,{\mathbb{S}}^3)$ and $d_{\mathrm{GH}}({\mathbb{S}}^2,{\mathbb{S}}^3)$. We also formulate a number of open questions.

* We made some structural changes for better readability

References in corpus (3)

• Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
• Torus cannot collapse to a segment
• Gromov-Hausdorff Distance Between Segment and Circle

Cited by in corpus (3)

• Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
• The Persistent Topology of Optimal Transport Based Metric Thickenings
• Geometric Bounds for Persistence