Translational tilings by a polytope, with multiplicity

TL;DR

This paper explores the geometric conditions under which convex bodies can cover Euclidean space exactly k times through translations, extending classical tiling results to multiple coverings.

Contribution

It generalizes Minkowski's classical tiling conditions to k-tilings, establishing symmetry properties necessary and sufficient for such coverings.

Findings

Convex bodies that k-tile are centrally symmetric.

Facets of k-tiling polytopes are centrally symmetric.

For rational polytopes, symmetry implies the existence of a k-tiling.

Abstract

We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d 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 and Minkowski. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R^d 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 R^d for some positive integer k.