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 $\mathbb{E}^3$.