Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings

The structural characterization of lattice coverings is a fundamental problem in the geometry of numbers. In particular, a covering analogue to Minkowski’s criterion for lattice packings has remained open. In this paper, we introduce the concept of $\mathit{\text{minimal}}\mathit{\text{covering}}\mathit{\text{bodies}}$ and investigate their structural properties. First, we establish a lattice covering criterion in three dimensions based on the Kuhn triangulation. Furthermore, while three-dimensional parallelohedra admit only finitely many combinatorial types, we prove the existence of infinitely many combinatorial types of minimal covering bodies in both the three-dimensional asymmetric case and the four-dimensional centrally symmetric case. Finally, we propose a Minkowski-type geometric criterion and an algebraic intersection framework, which reduce the three-dimensional covering problem to a finite computational verification.