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

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 $l_3^p$-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 $0.3840\dots$ to $0.3745\dots$, getting closer to the best known lower bound of $0.3673\dots$ 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.

30 pages, 6 tables, 3 figures, (v3) comments of referees incorporated

References in corpus (6)

• Symmetry groups, semidefinite programs, and sums of squares
• The sphere packing problem in dimension 24
• The sphere packing problem in dimension 8
• Upper bound on the packing density of regular tetrahedra and octahedra
• Some properties of optimal functions for sphere packing in dimensions 8 and 24
• The packing density of the $n$-dimensional cross-polytope

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• Solving clustered low-rank semidefinite programs arising from polynomial optimization
• New dense superball packings in three dimensions
• Semidefinite programming bounds for error-correcting codes