The Gromov–Hausdorff distance between spheres

Lim, Sunhyuk; Mémoli, Facundo; Smith, Zane

doi:10.2140/gt.2023.27.3733

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The Gromov–Hausdorff distance between spheres

Sunhyuk Lim, Facundo Mémoli and Zane Smith

Geometry & Topology 27 (2023) 3733–3800

DOI: 10.2140/gt.2023.27.3733

Abstract

We provide general upper and lower bounds for the Gromov–Hausdorff distance $d_{GH} (𝕊^{m}, 𝕊^{n})$ between spheres $𝕊^{m}$ and $𝕊^{n}$ (endowed with the round metric) for $0 \leq m < n \leq \infty$ . 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 $d_{GH} (𝕊^{0}, 𝕊^{n})$ , $d_{GH} (𝕊^{m}, 𝕊^{\infty})$ , $d_{GH} (𝕊^{1}, 𝕊^{2})$ , $d_{GH} (𝕊^{1}, 𝕊^{3})$ and $d_{GH} (𝕊^{2}, 𝕊^{3})$ . We also formulate a number of open questions.

Keywords

Gromov–Hausdorff distance, Borsuk–Ulam theorem

Mathematical Subject Classification

Primary: 53C23

References

Publication

Received: 7 September 2021

Revised: 7 April 2022

Accepted: 27 May 2022

Published: 5 December 2023

Proposed: Dmitri Burago

Seconded: Dan Abramovich, David Fisher

Authors

Sunhyuk Lim
Max Planck Institute for Mathematics in the Sciences Leipzig Germany
https://sites.google.com/view/sunhyuklim
Facundo Mémoli
Department of Mathematics The Ohio State University Columbus, OH United States
http://facundo-memoli.org
Zane Smith
Department of Computer Science University of Minnesota Minneapolis, MN United States

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Volume 27, issue 9 (2023)