Papers › The Gromov-Hausdorff distance between spheres

The Gromov-Hausdorff distance between spheres

3 May 2021arXiv:2105.00611links table onlyarchive 2025-07-28

Sunhyuk Lim, Facundo Mémoli, Zane Smith

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We provide general upper and lower bounds for the Gromov-Hausdorff distance d_(GH)(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 d_(GH)(𝕊⁰,𝕊ⁿ), d_(GH)(𝕊ᵐ,𝕊^∞), d_(GH)(𝕊¹,𝕊²), d_(GH)(𝕊¹,𝕊³) and d_(GH)(𝕊²,𝕊³). We also formulate a number of open questions.

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