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

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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.