Angle Sums of Schläfli Orthoschemes

Abstract

AbstractWe consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and $$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$ K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 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 sums of the conic intrinsic volumes of these tangent cones at all j-faces of $$K_n^A$$ K n A and $$K_n^B$$ K n B . This setting contains sums of external and internal angles of $$K_n^A$$ K n A and $$K_n^B$$ 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.