论文标题
简单多面体的角度总和
Angle sums of simplicial polytopes
论文作者
论文摘要
多层人士的内部角度矢量（$ \wideHatα$ - 矢量）是$ f $ - 矢量的度量类似物，其中面部面部由它们的固体角度加权。对于简单的多型，Sommerville（1927）和Höhn（1953）引入了$ \widehatα$ - vector上的Dehn-Sommerville型关系。 Camenga（2006）定义了$ \wideHatγ$ - 向量，这是一种类似于$ h $ vector的线性转换，并猜想它是非负的。使用几何和代数组合制剂中的工具，我们证明了这一猜想，并表明$ \wideHatγ$ - 矢量在上半年增加，并且是完美无瑕的。与$ h $ - 矢量相反，我们构建了一个六维多层，其非偶像$ \widehatγ$ - vector。更一般而言，当固体角度被简单和非负锥估值取代时，所有结果均保持有效。
The interior angle vector ($\widehatα$-vector) of a polytope is a metric analogue of the $f$-vector in which faces are weighted by their solid angle. For simplicial polytopes, Dehn-Sommerville-type relations on the $\widehatα$-vector were introduced by Sommerville (1927) and Höhn (1953). Camenga (2006) defined the $\widehatγ$-vector, a linear transformation analogous to the $h$-vector and conjectured it to be non-negative. Using tools from geometric and algebraic combinatorics, we prove this conjecture and show that the $\widehatγ$-vector increases in the first half and is flawless. In contrast to the $h$-vector, we construct a six-dimensional polytope with non-unimodal $\widehatγ$-vector. More generally, all result remain valid when solid angles are replaced by simple and non-negative cone valuations.
