Optimality and uniqueness of the D₄ root system

2024/04/29 by David de Laat, de Laat, David, Nando Leijenhorst +3 · 2 citations
Engineering · #52C17 #90C22 #Advanced Control Systems Design #Advanced Control Systems Optimization #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2404.18794

openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the D₄ root system (the set of vertices of the regular 24-cell) is the unique optimal kissing configuration in \mathbb R⁴, 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 R⁶ to 77.

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• Conic optimization for extremal geometry
• The Augmented Mixing Method: Computing High-Accuracy Primal-Dual Solutions to Large-Scale SDPs via Column Updates

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