Minkowski-type criterion for three-dimensional lattice coverings

Let $K\subseteq\mathbb{R}^3$ be an origin-symmetric convex body. Let $\Delta_1,\dots,\Delta_6$ be the six standard Kuhn simplices in $\mathbb{R}^3$.

Minkowski-type criterion for three-dimensional lattice coverings. The arrangement $K+\mathbb{Z}^3$ is a lattice covering of $\mathbb{R}^3$ if and only if there exist a unimodular transformation $U\in\operatorname{GL}(3,\mathbb{Z})$ and translation vectors $\mathbf{x}_1,\dots,\mathbf{x}_6\in\mathbb{R}^3$ such that

$$U(\Delta_i)+\mathbf{x}_i\subseteq K\quad\text{for all }i=1,\dots,6.$$

This is presented as the covering analogue of Minkowski's criterion for optimal three-dimensional lattice packings. Its validity depends on the preceding proposed classification of centrally symmetric minimal covering bodies.