Angle Sums of Schläfli Orthoschemes

Abstract

We consider the simplices KnA={x∈Rn+1:x1≥x2≥⋯≥xn+1,x1-xn+1≤1,x1+⋯+xn+1=0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n^A=\\{x\\in {\\mathbb {R}}^{n+1}:x_1\\ge x_2\\ge \\cdots \\ge x_{n+1},x_1-x_{n+1}\\le 1,\\,x_1+\\cdots +x_{n+1}=0\\} \\end{aligned}$$\\end{document}and KnB={x∈Rn:1≥x1≥x2≥⋯≥xn≥0},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n^B=\\{x\\in {\\mathbb {R}}^n:1\\ge x_1\\ge x_2\\ge \\cdots \\ge x_n\\ge 0\\}, \\end{aligned}$$\\end{document}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.