Kissing number in spherical space

TL;DR

This paper explores how the maximum number of non-overlapping equal spheres that can touch a central sphere in spherical space varies with radius, providing high-accuracy bounds using advanced optimization and coding techniques.

Contribution

It introduces new upper and lower bounds for the kissing number in spherical space that depend on the sphere radius, improving understanding of this geometric problem.

Findings

Graph of kissing number versus radius with high accuracy

New bounds derived via semidefinite programming and spherical codes

Enhanced understanding of sphere packings in spherical geometry

Abstract

This paper investigates the behaviour of the kissing number $\kappa(n, r)$ of congruent radius $r > 0$ spheres in $\mathbb{S}^n$, for $n\geq 2$. Such a quantity depends on the radius $r$, and we plot the approximate graph of $\kappa(n, r)$ with relatively high accuracy by using new upper and lower bounds that are produced via semidefinite programming and by using spherical codes, respectively.