多胞形的谱性与平移可等分解性
Spectrality of polytopes and equidecomposability by translations
摘要：
设$A$是$\mathbb{R}^d$中的一个多面体（不一定是凸的或连通的）。如果空间$L^2(A)$有一个由指数函数组成的正交基，我们称$A$是谱的。Kolountzakis和Papadimitrakis（2002）的一个结果断言，如果$A$是一个谱多面体，那么向外法线指向给定方向的$A$的$(d-1)$维面的总面积必须与向外法线指向相反方向的$(d-1)$维面的总面积一致。在本文中，我们证明了这一结果的扩展$1$和$d-1$之间的所有维度的面。因此，我们得到任何频谱多面体$A$可以被分割成一个有限数量的较小的多面体，可以重新排列使用的翻译，以形成一个立方体。
英文摘要：
Let $A$ be a polytope in $\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis (2002) asserts that if $A$ is a spectral polytope, then the total area of the $(d-1)$-dimensional faces of $A$ on which the outward normal is pointing at a given direction, must coincide with the total area of those $(d-1)$-dimensional faces on which the outward normal is pointing at the opposite direction. In this paper, we prove an extension of this result to faces of all dimensions between $1$ and $d-1$. As a consequence we obtain that any spectral polytope $A$ can be dissected into a finite number of smaller polytopes, which can be rearranged using translations to form a cube.
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