Translational tilings by a polytope, with multiplicity |
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Authors: | Nick Gravin Sinai Robins Dmitry Shiryaev |
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Institution: | 1. Division of Mathematical Sciences, Nanyang Technological University, SPMS-MAS-03-01 21 Nanyang Link, Singapore, 637371, Singapore
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Abstract: | We study the problem of covering ? d by overlapping translates of a convex polytope, such that almost every point of ? d is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov 5] and Minkowski 15], and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile ? d . By contrast, for k ≥2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles ? 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, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational polytope, is centrally symmetric, and has centrally symmetric facets, then P must k-tile ? d for some positive integer k. |
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