Absorption probabilities for Gaussian polytopes and regular spherical simplices. (June 2020) - Kabluchko, Zakhar; Zaporozhets, Dmitry
Abstract: The Gaussian polytope $\mathcal P_{n, d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n, d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n, d}$ contains the point $σ X$, where $σ≥ 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n, d}$ . As a by-product, we also compute the expected number of k -faces and the expected volume of $\mathcal P_{n, d}$, 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.