Translational tilings by a polytope, with multiplicity

We study the problem of covering Rd by overlapping translates of a convex body P, such that almost every point of Rd is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov [3] and Minkowski [11]. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles Rd by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes. Conversely, in the case that P is a rational polytope, we also prove that if P is centrally symmetric and has centrally symmetric facets, then P must k-tile Rd for some positive integer k