Optimality and uniqueness of the $D_4$ root system

TL;DR

This paper proves the uniqueness and optimality of the $D_4$ root system as a kissing configuration in four-dimensional space and improves bounds for the kissing number in six dimensions using advanced optimization techniques.

Contribution

It establishes the $D_4$ root system as the unique optimal configuration in $\, ext{R}^4$ and enhances the upper bound for the kissing number in $\, ext{R}^6$ with semidefinite programming.

Findings

$D_4$ root system is the unique optimal kissing configuration in $\, ext{R}^4$

Exact optimal solution computed via semidefinite programming

Upper bound for kissing number in $\, ext{R}^6$ improved to 77

Abstract

We prove that the $D_4$ root system (the set of vertices of the regular $24$-cell) is the unique optimal kissing configuration in $\mathbb R^4$, and is an optimal spherical code. For this, we use semidefinite programming to compute an exact optimal solution to the second level of the Lasserre hierarchy. We also improve the upper bound for the kissing number problem in $\mathbb R^6$ to $77$.