Angle sums of Schl\"afli orthoschemes

Abstract

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 Schl\"afli 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 Schl\"afli 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.