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

Abstract

The precise structural characterization of lattice coverings remains a profound open problem in the geometry of numbers, notably lacking an analogue to Minkowski's classical criterion for lattice packings. To bridge this gap, we introduce and structurally classify a novel geometric framework: \textit{minimal covering bodies}. First, we establish a constructive lattice covering criterion in three dimensions based on the classical Kuhn triangulation. Furthermore, while classical space-tiling parallelohedra are limited to finitely many combinatorial types, we prove that minimal covering bodies can exhibit infinitely many distinct types in both asymmetric and symmetric higher-dimensional settings. Finally, we propose a Minkowski-type geometric criterion and an algebraic intersection framework, aiming to reduce the continuous three-dimensional covering problem to a computationally verifiable discrete condition.