{"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/kissing-number-in-non-euclidean-spaces","title":"Kissing number in non-Euclidean spaces of constant sectional curvature","arxiv_id":"2003.05547","date":"2020-03-11","proceeding":null,"authors":["Maria Dostert","Alexander Kolpakov"],"abstract":"This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in hyperbolic $\\mathbb{H}^n$ and spherical $\\mathbb{S}^n$ spaces, for $n\\geq 2$. For that purpose, the kissing number is replaced by the kissing function $\\kappa_H(n, r)$, resp. $\\kappa_S(n, r)$, which depends on the dimension $n$ and the radius $r$. After we obtain some theoretical upper and lower bounds for $\\kappa_H(n, r)$, we study their asymptotic behaviour and show, in particular, that $\\kappa_H(n,r) \\sim (n-1) \\cdot d_{n-1} \\cdot B(\\frac{n-1}{2}, \\frac{1}{2}) \\cdot e^{(n-1) r}$, where $d_n$ is the sphere packing density in $\\mathbb{R}^n$, and $B$ is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of $\\kappa_S(n, r)$, for $n= 3,\\, 4$, over subintervals in $[0, \\pi]$ with relatively high accuracy.","url_abs":"https://arxiv.org/abs/2003.05547v6","url_pdf":"https://arxiv.org/pdf/2003.05547v6.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":"kissing-number-in-non-euclidean-spaces","repo_url":"https://github.com/sashakolpakov/non-euclidean-kissing-number","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}