Kissing number in non-Euclidean spaces of constant sectional curvature

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 $n\ge 2$. 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.

17 pages, 2 figures, 4 tables; ancillary files available on GitHub: https://github.com/sashakolpakov/non-euclidean-kissing-number This is a merger of arXiv:1907.00255 and arXiv:1910.02715 (both withdrawn)

References in corpus (2)

• New upper bounds for kissing numbers from semidefinite programming
• Contact graphs of ball packings