凸多面体Minkowski和上界定理的几何方法
A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes
摘要：
我们推导出紧表达式的最大数目的$k$ -面，$0\le{}k\le{}d-1$，闵可夫斯基和，$P_1+...+P_r$，$r$凸$d$ -多面体$P_1,...,P_r$在$\mathbb{R}^d$，其中$d\ge{}2$和$r<d$，作为一个（递归定义的）函数的顶点数的多面体。我们的结果与Adiprasito和Sanyal最近证明的结果一致。与Adiprasito和Sanyal使用组合交换代数工具的方法相反，我们的方法是纯几何的，并使用基本概念，如$f$ -和$h$ -向量微积分和壳，并推广了[15]和[14]中用于证明两个和三个凸多面体的Minkowski和的$f$ -向量上界的方法。我们的方法背后的关键思想是将Minkowski和$P_1+...+P_r$表示为被加数的Cayley多胞形$\mathcal{C}$的一部分;对$P_1+...+P_r$的$k$ -面进行绑定简化为对包含来自每个$r$多胞形的顶点的$\mathcal{C}$的$(k+r-1)$ -面的子集进行绑定。我们结束我们的文件与草图的明确的建设，建立了紧密的上界。
英文摘要：
We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex $d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ and $r<d$, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [2]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as $f$- and $h$-vector calculus and shellings, and generalizes the methodology used in [15] and [14] for proving upper bounds on the $f$-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum $P_1+...+P_r$ as a section of the Cayley polytope $\mathcal{C}$ of the summands; bounding the $k$-faces of $P_1+...+P_r$ reduces to bounding the subset of the $(k+r-1)$-faces of $\mathcal{C}$ that contain vertices from each of the $r$ polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.
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