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The Gromov–Hausdorff distance between spheres

Sunhyuk Lim, Facundo Mémoli and Zane Smith

Geometry & Topology 27 (2023) 3733–3800
Abstract

We provide general upper and lower bounds for the Gromov–Hausdorff distance dGH (𝕊m, 𝕊n) between spheres 𝕊m and 𝕊n (endowed with the round metric) for 0 m < n . 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 dGH (𝕊0, 𝕊n), dGH (𝕊m, 𝕊), dGH (𝕊1, 𝕊2), dGH (𝕊1, 𝕊3) and dGH (𝕊2, 𝕊3). 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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