Translational tilings by a polytope, with multiplicity
Translational tilings by a polytope, with multiplicity
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具有多重性的多面体平移平铺
DOI：
10.1007/s00493-012-2860-3
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发表时间：
2011
期刊：
影响因子：
1.1
通讯作者：
D. Shiryaev
中科院分区：
文献类型：
--
作者：
N. Gravin;S. Robins;D. Shiryaev
我们研究了通过凸多面体的重叠平移来覆盖 ℤd 的问题，使得 ℤd 的几乎每个点都被精确地覆盖了 k 次。这种通过一组离散的平移来覆盖欧几里得空间的方法称为 k-平铺。通过平移对简单平铺（在本文中我们称为 1-平铺）的研究始于 Fedorov [5] 和 Minkowski [15] 的工作，后来由 Venkov 和 McMullen 扩展，给出了 1-tile ℤd 的所有凸对象的完整表征。相比之下，对于 k ≥2，k-tile 的多胞体集合比 1-tile 的多胞体集合宽得多，并且有目前还没有已知的 k-tile 多胞体的类似表征。这里我们首先给出多面体 P 即 k-tile 的必要条件，通过证明如果 P k-tiles ℤd 通过平移，那么它是中心对称的，并且它的面也是中心对称的。这些与 Minkowski 的 1-平铺多胞体条件类似，但事实证明，该理论的发展需要非常新的方法。在 P 有有理顶点的情况下，我们也证明逆命题成立；也就是说，如果 P 是有理多面体，是中心对称的，并且具有中心对称面，那么对于某个正整数 k，P 必须 k-tile ℤd。
We study the problem of covering ℤd by overlapping translates of a convex polytope, such that almost every point of ℤd 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 ℤd.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 ℤd 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 ℤd for some positive integer k.