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. These bounds give strong indications that some of the lattice packings of superballs found in 2009 by Jiao, Stillinger, and Torquato are indeed optimal among all translative packings. 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 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.

Additional Metadata
Keywords Geometric packings, Density, Upper Bounds, Polynomial optimization
THEME Null option (theme 11)
Publisher Cornell University Library
Series e-Print archive
Dostert, M, Guzman Paredes, C.A, de Oliveira Filho, F.M, & Vallentin, F. (2015). New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry. e-Print archive. Cornell University Library .