The Simultaneous Lattice Packing-Covering Constant of Octahedra

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