New upper bounds for average kissing numbers
A packing of balls in Euclidean space is a set of balls with non-overlapping interiors. For a finite packing of balls, its average contact number is the average number of balls tangent to a ball in this packing, that is, twice the total number of tangencies divided by the number of balls. The average kissing number of the space is defined as the supremum of average contact numbers over all finite ball packings. In this paper, we present a new semidefinite programming approach to average kissing numbers and improve upon known upper bounds in all dimensions \(3\) through \(30\). In particular, we provide the first improvements to the upper bounds in dimensions \(10\) through \(30\) obtained from kissing numbers. We also fully solve the average kissing number problem in dimensions \(8\) and \(24\) by showing that the average kissing numbers coincide with the kissing numbers, 240 and 196560, respectively. These values are limits of average contact numbers of finite subpackings from the \(E_8\) and Leech lattices.