大量多面体的Minkowski和的极大f-向量
Maximal f-vectors of Minkowski sums of large numbers of polytopes
摘要：
已知在$d$维中的$r$多面体的Minkowski和中，其中$r<d$，和的顶点数可能与每个被加数中的顶点数的乘积一样高。然而，到目前为止，多个多面体之和的顶点数是未知的。本文研究了一般方向上的多面体的和，证明了维数为$d$的$r$多面体的和的面数与维数小于$d$的被加数多面体的和的面数之间的线性关系.我们从这个确切的公式推导出一个紧界上的最大可能的顶点数的闵可夫斯基和的任何数量的多面体在任何维度。特别地，该线性关系意味着维度$d$中的$r$多面体的和具有被加数中的顶点总数的$O(n^{d-1})$中的顶点数，即使当$r\geq d$。这个界是紧的，因为某些和确实有那么多顶点。
英文摘要：
It is known that in the Minkowski sum of $r$ polytopes in dimension $d$, with $r<d$, the number of vertices of the sum can potentially be as high as the product of the number of vertices in each summand. However, the number of vertices for sums of more polytopes was unknown so far. In this paper, we study sums of polytopes in general orientations, and show a linear relation between the number of faces of a sum of $r$ polytopes in dimension $d$, with $r\geq d$, and the number of faces in the sums of less than $d$ of the summand polytopes. We deduce from this exact formula a tight bound on the maximum possible number of vertices of the Minkowski sum of any number of polytopes in any dimension. In particular, the linear relation implies that a sum of $r$ polytopes in dimension $d$ has a number of vertices in $O(n^{d-1})$ of the total number of vertices in the summands, even when $r\geq d$. This bound is tight, in the sense that some sums do have that many vertices.
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