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

Robins, Sinai

### 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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