Algebraic intersection criterion for symmetric lattice coverings

Let $K\subseteq\mathbb{R}^3$ be an origin-symmetric convex body, let $\lambda>0$, and let $v_{i,1},\dots,v_{i,4}$ be the vertices of the six standard Kuhn simplices $\Delta_i$.

Algebraic intersection criterion. The arrangement $\lambda K+\mathbb{Z}^3$ is a lattice covering of $\mathbb{R}^3$ if and only if there exists a unimodular transformation $U\in\operatorname{GL}(3,\mathbb{Z})$ such that, for every $i=1,\dots,6$,

$$\bigcap_{l=1}^4\bigl(\lambda K+U(v_{i,l})\bigr)\neq\varnothing.$$

Consequently, the covering radius $\rho(K)$ is the minimal $\lambda>0$ satisfying this intersection condition. The criterion is intended to eliminate the translation variables from the preceding geometric formulation and make the covering problem computationally tractable.