Algorithmic verification criterion for the restricted asymmetric case

Let $K\subseteq\mathbb{R}^3$ be a not necessarily symmetric convex body. Suppose a specific combinatorial type of Delone decomposition is generated by primitive lattice polytopes $P_1,\dots,P_k$. For a scaling factor $\lambda>0$, let $v_{j,1},\dots,v_{j,m_j}$ be the vertices of $P_j$.

Algorithmic verification criterion. The following conditions are equivalent: the configuration $\lambda K+\mathbb{Z}^3$ is a lattice covering realizing the specified combinatorial type; there exist a unimodular transformation $U\in\operatorname{GL}(3,\mathbb{Z})$ and translations $\mathbf{x}_1,\dots,\mathbf{x}_k\in\mathbb{R}^3$ with

$$U(P_j)+\mathbf{x}_j\subseteq\lambda K\quad\text{for all }j=1,\dots,k;$$

and there exists a unimodular transformation $U\in\operatorname{GL}(3,\mathbb{Z})$ such that, for every $j=1,\dots,k$,

$$\bigcap_{l=1}^{m_j}\bigl(\lambda K-U(v_{j,l})\bigr)\neq\varnothing.$$

The criterion is proposed within the restricted asymmetric classification framework, where bounding the volumes of primitive Delone polytopes reduces the possible combinatorial types to a finite set and permits computational verification.