Solid-angle formulas for simplicial cones and random shifted sections of polytopes

Abstract

We consider simplicial cones $C = \operatorname{pos}(v_1, \ldots, v_n) \subseteq \mathbb{R}^n$ whose generators have a Gram matrix of the form $(\langle v_i, v_j \rangle)_{i,j=1}^n = \operatorname{Id}_n+\sum_{j=1}^p λ_j u_j u_j^T$, where $p\in \{0,\ldots, n\}$, $λ_1, \ldots, λ_p > -1$ and $u_1,\ldots, u_p \in \mathbb{R}^n$ are orthogonal unit vectors. We derive an explicit $p$-dimensional integral formula for the solid angle of $C$ involving the analytic continuation of the standard normal distribution function. As an application, explicit formulas are obtained for the expected face numbers of sections of acute orthocentric simplices and crosspolytopes with uniformly distributed linear subspaces translated by a fixed vector. The translation vector is assumed to lie outside the affine hulls of all proper faces of the polytope. All solid angles entering these formulas are expressed as integrals of dimension at most two. This application includes regular simplices and regular crosspolytopes as special cases.