TY - JOUR
T1 - New Upper Bounds for the Density of Translative Packings of Three-Dimensional Convex Bodies with Tetrahedral Symmetry
AU - Dostert, Maria
AU - Guzmán Paredes, Cristóbal Andrés
AU - Filho, Fernando Mário de Oliveira
AU - Vallentin, Frank
PY - 2017/3/9
Y1 - 2017/3/9
N2 - 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.
AB - 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.
U2 - 10.1007/s00454-017-9882-y
DO - 10.1007/s00454-017-9882-y
M3 - Article
SN - 0179-5376
VL - 58
SP - 449
EP - 481
JO - Discrete and computational geometry
JF - Discrete and computational geometry
ER -