Non-linear angle-sum relations for polyhedral cones and polytopes
Non-linear angle-sum relations for polyhedral cones and polytopes
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多面体锥体和多面体的非线性角和关系
DOI：
10.1017/s0305004100051665
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发表时间：
1975
影响因子：
0.8
通讯作者：
P. McMullen
中科院分区：
文献类型：
--
作者：
P. McMullen
摘要研究了多面体锥面的内角和外角满足各种双线性关系。前两个公式与球面多面体的Gauss-Bonnet和Steiner平行公式有关，而第三个公式则是全新的。然而，这些证明本质上基本上是组合的，而不是像更经典的方法那样是微分几何的。对于定义在多面体和多面体锥上的某些函数，这些关系导致了类似于欧拉型关系的反演公式。结果发现了涉及到quermassintegral和Grassmann角的各种新的关系；对晶格多面体也有一个应用。
Abstract It is shown that the internal and external angles at the faces of a polyhedral cone satisfy various bilinear relations. The first two of these are related to the Gauss–Bonnet and Steiner parallel formulae for spherical polytopes, while the third is completely new. However, the proofs are basically combinatorial in nature, rather than differential geometric, as in the more classical treatments. These relations lead to inversion formulae, analogous to Euler-type relations, for certain functions defined on polytopes and polyhedral cones. As a result, various new relations involving quermassintegrals and Grassmann angles are found; there is also an application to lattice polytopes.