Discrete & Computational Geometry ( IF 0.6 ) Pub Date : 2021-10-05 , DOI: 10.1007/s00454-021-00326-z Thomas Godland 1 , Zakhar Kabluchko 1
We consider the simplices
$$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$and
$$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$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 sums 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.
中文翻译：
Schläfli Orthoschemes 的角和
我们考虑简单的
$$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_ {n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$和
$$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned }$$分别称为A 型和 B 型的Schläfli 正畸方案。我们在它们的j面上描述切锥并明确计算这些切锥在\(K_n^A\)和 \(K_n^B\) 的所有j面上的圆锥内在体积的总和。此设置包含\(K_n^A\)和\(K_n^B\) 的外角和内角之和作为特殊情况。总和根据两种类型的斯特林数进行评估。我们将这些结果推广到A 类和B类 Schläfli 正畸方案的有限乘积，作为概率结果，推导出j的预期数量的公式-有限数量的高斯随机游走和随机桥的凸包的闵可夫斯基和的面。此外，我们评估了A 型和B型外尔室的切锥及其有限乘积的类似角和。
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