Kissing number in non-Euclidean spaces of constant sectional curvature

Author:

Dostert Maria,Kolpakov Alexander

Abstract

This paper provides upper and lower bounds on the kissing number of congruent radius r > 0 r > 0 spheres in hyperbolic H n \mathbb {H}^n and spherical S n \mathbb {S}^n spaces, for n 2 n\geq 2 . For that purpose, the kissing number is replaced by the kissing function κ H ( n , r ) \kappa _H(n, r) , resp. κ S ( n , r ) \kappa _S(n, r) , which depends on the dimension n n and the radius r r .

After we obtain some theoretical upper and lower bounds for κ H ( n , r ) \kappa _H(n, r) , we study their asymptotic behaviour and show, in particular, that κ H ( n , r ) ( n 1 ) d n 1 B ( n 1 2 , 1 2 ) e ( n 1 ) r \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 d_n is the sphere packing density in R n \mathbb {R}^n , and B 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 κ S ( n , r ) \kappa _S(n, r) , for n = 3 , 4 n= 3,\, 4 , over subintervals in [ 0 , π ] [0, \pi ] with relatively high accuracy.

Funder

Knut och Alice Wallenbergs Stiftelse

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

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