Absorption probabilities for Gaussian polytopes and regular spherical simplices. (June 2020)

• Record Type: Journal Article
• Authors: Kabluchko, Zakhar; Zaporozhets, Dmitry
• Is Part Of: Advances in applied probability. Volume 52:Number 2(2020)
• Page Start: 588
• Page End: 616
• Publication Date: 2020-06
• Subjects: Convex hull, -- random polytope, -- Gaussian polytope, -- Goodman–Pollack model, -- absorption probability, -- Wendel's formula, -- regular simplex, -- spherical geometry, -- solid angle, -- convex cone, -- conic Crofton formula, -- average number of faces, -- error function, -- Schläfli's function; 60D05, -- 52A22, -- 52B11, -- 60G70, -- 52A20, -- 52A23; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2020.7
• Languages: English
• ISSNs: 0001-8678

Abstract

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 $\sigma X$, where $\sigma\geq 0$ 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 $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.