Absorption probabilities for Gaussian polytopes and regular spherical simplices. (June 2020) - Kabluchko, Zakhar; Zaporozhets, Dmitry
Abstract: The Gaussian polytope P_n, d is the convex hull of n independent standard normally distributed points in R^d . We derive explicit expressions for the probability that P_n, d contains a fixed point xR^d as a function of the Euclidean norm of x, and the probability that P_n, d contains the point σ X, where σ≥ 0 is constant and X is a standard normal vector independent of P_n, d . As a by-product, we also compute the expected number of k -faces and the expected volume of P_n, d, thus recovering the results of Affentranger and Schneider ( Discr. and Comput. Geometry, 1992) and Efron ( Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function Φ(z) and its complex version Φ(iz) . The main tool used in the proofs is the conic version of the Crofton formula.