The Gromov–Hausdorff distance between spheres

Lim, Sunhyuk¹ ; Mémoli, Facundo² ; Smith, Zane³

¹ Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
² Department of Mathematics, The Ohio State University, Columbus, OH, United States
³ Department of Computer Science, University of Minnesota, Minneapolis, MN, United States

Geometry & topology, Tome 27 (2023) no. 9, pp. 3733-3800.

Résumé

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.

Détail

DOI : 10.2140/gt.2023.27.3733

Keywords: Gromov–Hausdorff distance, Borsuk–Ulam theorem

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