New Upper Bounds for the Density of Translative Packings of Three-Dimensional Convex Bodies with Tetrahedral Symmetry

Maria Dostert, Cristóbal Andrés Guzmán Paredes, Fernando Mário de Oliveira Filho, Frank Vallentin

Research output: Contribution to journalArticleAcademicpeer-review

8 Citations (Scopus)

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 lp3-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… to 0.3745…, getting closer to the best known lower bound of 0.3673… 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.
Original languageEnglish
Pages (from-to)449-481
JournalDiscrete and computational geometry
Volume58
DOIs
Publication statusPublished - 9 Mar 2017
Externally publishedYes

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