High accuracy semidefinite programming bounds for kissing numbers

The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for n <= 24. The bound for n = 16 implies a conjecture of Conway and Sloane: There is no 16-dimensional periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5 + 61440q^6 + ...Comment: 7 pages (v3) new numerical result in Section 4, to appear in Experiment. Mat

A short solution of the kissing number problem in dimension three

In this note, we give a short solution of the kissing number problem in dimension three

Bounds for solid angles of lattices of rank three

We find sharp absolute constants $C_1$ and $C_2$ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval $[C_1,C_2]$. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in $\mathbb R^N$ with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in $\mathbb R^3$. Such spherical configurations come up in connection with the kissing number problem.Comment: 12 pages; to appear in the Journal of Combinatorial Theory

On kissing numbers and spherical codes in high dimensions

We prove a lower bound of $\Omega (d^{3/2} \cdot (2/\sqrt{3})^d)$ on the kissing number in dimension $d$. This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle $\theta$ in high dimensions

Contact graphs of ball packings

A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $\mathbb{R}^3$ is not greater than $13.955$. We also find new upper bounds for the average degree of contact graphs in $\mathbb{R}^4$ and $\mathbb{R}^5$

The strong thirteen spheres problem

The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag