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

arXiv: https://arxiv.org/abs/1510.02331 · HTML: https://mlsanity.com/paper/1510.02331/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Oct 2015.

## Built on by (most cited first, 1 of 1)

- [Sparse Polynomial Optimization: Theory and Practice](https://mlsanity.com/paper/2208.11158/) (2022)

## Abstract (arXiv)

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 0.3840ldots to 0.3745ldots, getting closer to the best known lower bound of 0.3673ldots 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.