The Simultaneous Lattice Packing-Covering Constant of Octahedra

Abstract

This paper proves that the simultaneous lattice packing-covering constant of an octahedron is $7/6$. In other words, $7/6$ is the smallest positive number $r$ such that for every octahedron $O$ centered at the origin there is a lattice $\Lambda$ such that $O+\Lambda$ is a packing in ${\mathbb{E}}^3$ and $rO+\Lambda$ is a covering of ${\mathbb{E}}^3$.

24 pages, 3 figures