Angle Sums of Schläfli Orthoschemes

Abstract We consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = { x ∈ R + 1 : ≥ 2 ⋯ , - ≤ 0 } and K_n^B=\{x\in {R}}^n:1\ge x_1\ge x_n\ge 0\}, B which are called Schläfli orthoschemes of types A B , respectively. describe tangent cones at their j -faces compute explicitly sums conic intrinsic volumes these all $$K_n^A$$ $$K_n^B$$ . This setting contains external internal angles as special cases. The evaluated in terms Stirling numbers both kinds. generalize results to finite products type and, a probabilistic consequence, derive formulas for expected number Minkowski convex hulls Gaussian random walks bridges. Furthermore, we evaluate analogous angle Weyl chambers thereof.