三个凸多面体的Minkowski和的最大面数
The maximum number of faces of the Minkowski sum of three convex polytopes
摘要：
我们导出了三个$d$ -维凸多面体$P_1$，$P_2$和$P_3$的Minkowski和的最大$k$ -面数$0\le k\le d-1$，$P_1+P_2+P_3$的紧表达式，作为多面体顶点数的函数，对于任何$d\ge 2$ .将三个多面体的Minkowski和表示为它们的Cayley多面体$\mathcal{C}$的一部分，计算$P_1+P_2+P_3$的$k$ -面的数目的问题减少到计算由包含来自每个$P_i$的至少一个顶点的面组成的$\mathcal{C}$的子集的$(k+2)$ -面的数目。在两个维度，我们的表达式减少到已知的结果，而在三个维度，紧接着利用已知的紧边界$r$ $d$ -多面体的面数，其中$r\ge d$我们的边界。对于$d\ge 4$，当$P_1$，$P_2$和$P_3$是$d$ -多面体时，其顶点集从三个不同的$d$ -维类矩曲线中适当选取，得到最大值.
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, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the polytopes, for any $d\ge 2$. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope $\mathcal{C}$, the problem of counting the number of $k$-faces of $P_1+P_2+P_3$, reduces to counting the number of $(k+2)$-faces of the subset of $\mathcal{C}$ comprising of the faces that contain at least one vertex from each $P_i$. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of $r$ $d$-polytopes, where $r\ge d$. For $d\ge 4$, the maximum values are attained when $P_1$, $P_2$ and $P_3$ are $d$-polytopes, whose vertex sets are chosen appropriately from three distinct $d$-dimensional moment-like curves.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
