论文标题
schläfli矫形器的角度总和
Angle sums of Schläfli orthoschemes
论文作者
论文摘要
我们考虑简单$$ k_n^a = \ {x \ in \ mathbb {r}^{n+1}：x_1 \ ge x_2 \ ge x_2 \ ge \ ge \ ge x_ x_ x_ x_ {n+1}，x_1-x_ {x_1-x_ {n+1} {n+1} k_n^b = \ {x \ in \ mathbb {r}^n：1 \ ge x_1 \ ge x_1 \ ge x_2 \ ge \ ge \ ldots \ ge x_n \ ge 0 \}，$$，分别称为schläfliorthoschemes of类型$ a $ a $ a $ a $ a和$ b $。我们以$ j $ face的形式描述切线锥，并明确计算这些切线锥体的圆锥固有卷的总和，这些圆锥体的所有$ j $ - faces $ k_n^a $ and $ k_n^b $。此设置包含$ k_n^a $和$ k_n^b $的外部和外部角度的总和。总和是根据两种史密斯数量评估的。我们将这些结果推广到$ a $ a $ a $ a $ and $ b $的Schläfli矫正器的有限产品，并且作为概率的结果，它得出了预期的$ j $ faces $ j $ faces的公式，这些convex hulls的Minkowski总和是有限的高斯随机步行和随机步行和随机桥梁的。此外，我们评估了$ a $ a $ and $ b $的Weyl Chambers的切线和有限产品的类似角度总和。
We consider the simplices $$ K_n^A=\{x\in\mathbb{R}^{n+1}:x_1\ge x_2\ge \ldots\ge x_{n+1},x_1-x_{n+1}\le 1,x_1+\ldots+x_{n+1}=0\} $$ and $$ K_n^B=\{x\in\mathbb{R}^n:1\ge x_1\ge x_2\ge \ldots\ge x_n\ge 0\}, $$ which are called the Schläfli orthoschemes of types $A$ and $B$, respectively. We describe the tangent cones at their $j$-faces and compute explicitly the sum of the conic intrinsic volumes of these tangent cones at all $j$-faces of $K_n^A$ and $K_n^B$. This setting contains sums of external and internal angles of $K_n^A$ and $K_n^B$ as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type $A$ and $B$ and, as a probabilistic consequence, derive formulas for the expected number of $j$-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types $A$ and $B$ and finite products thereof.
