Translational tilings of Euclidean space by a convex body, with multiplicity

Abstract

We study the problem of covering TeX Embedding failed! by overlapping translates of a convex body TeX Embedding failed!, such that almost every point of TeX Embedding failed! is covered exactly TeX Embedding failed! times, for a fixed integer TeX Embedding failed!. Such a covering of Euclidean space by translations is called a TeX Embedding failed!-tiling. The traditional 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 Fedorov and Minkowski to TeX Embedding failed! tilings by proving that if a convex body TeX Embedding failed!-tiles TeX Embedding failed! by translations, then it is centrally symmetric, and its facets are also centrally symmetric. The methods are very new, and they give the analogues of Minkowski’s conditions for 1-tiling polytopes. Conversely, in the case that TeX Embedding failed! is a rational polytope, we also prove that if TeX Embedding failed! is centrally symmetric and has centrally symmetric facets, then TeX Embedding failed! must TeX Embedding failed!-tile TeX Embedding failed! for some positive integer TeX Embedding failed!.

This is joint work with Nick Gravin and Dmitry Shiryaev.

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