New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry

TL;DR

This paper establishes new, tighter upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry, using advanced polynomial optimization techniques.

Contribution

It introduces a novel application of invariant theory of pseudo-reflection groups to improve upper bounds for packing densities of tetrahedral symmetric bodies.

Findings

Improved upper bound for regular tetrahedra packing density to 0.3745

Extended bounds for superballs and Platonic/Archimedean solids with tetrahedral symmetry

Application of computational, rigorous methods based on polynomial optimization

Abstract

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.