{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-gromov-hausdorff-distance-between-spheres","title":"The Gromov-Hausdorff distance between spheres","arxiv_id":"2105.00611","date":"2021-05-03","proceeding":null,"authors":["Sunhyuk Lim","Facundo Mémoli","Zane Smith"],"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 $0\\leq m< n\\leq \\infty$. 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 $d_{\\mathrm{GH}}(\\mathbb{S}^0,\\mathbb{S}^n)$, $d_{\\mathrm{GH}}(\\mathbb{S}^m,\\mathbb{S}^\\infty)$, $d_{\\mathrm{GH}}(\\mathbb{S}^1,\\mathbb{S}^2)$, $d_{\\mathrm{GH}}(\\mathbb{S}^1,\\mathbb{S}^3)$ and $d_{\\mathrm{GH}}(\\mathbb{S}^2,\\mathbb{S}^3)$. We also formulate a number of open questions.","url_abs":"https://arxiv.org/abs/2105.00611v6","url_pdf":"https://arxiv.org/pdf/2105.00611v6.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-gromov-hausdorff-distance-between-spheres","repo_url":"https://github.com/ndag/dgh-spheres","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}