Angle sums of Schl\"afli orthoschemes

TL;DR

This paper computes angle sums of Schl"afli orthoschemes of types A and B, relating them to Stirling numbers, and applies these results to probabilistic models involving Gaussian random walks and Weyl chambers.

Contribution

It provides explicit formulas for angle sums of Schl"afli orthoschemes and their products, linking geometric and probabilistic properties in a novel way.

Findings

Explicit angle sum formulas in terms of Stirling numbers.

Extension to products of orthoschemes and Weyl chambers.

Probabilistic formulas for expected face counts of Minkowski sums.

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.