In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley.

中文翻译：

---

Stanley-Reisner相对理论和Minkowski和的上界定理

在本文中，我们解决了两个长期以来关于多边形的Minkowski和的组合复杂性的问题：我们给出了Minkowski和的面数的严格上限，包括对等式的刻画。同样，我们给出了Minkowski和的混合面的（紧）上限定理。这具有广泛的应用范围，并推广了McMullen和Stanley的经典上界定理。