TY - JOUR

T1 - Absorption probabilities for Gaussian polytopes and regular spherical simplices

AU - Kabluchko, Zakhar

AU - Zaporozhets, Dmitry

PY - 2020/6/1

Y1 - 2020/6/1

N2 - The Gaussian polytope is the convex hull of n independent standard normally distributed points in. We derive explicit expressions for the probability that contains a fixed point as a function of the Euclidean norm of x, and the probability that contains the point, where is constant and X is a standard normal vector independent of. As a by-product, we also compute the expected number of k-faces and the expected volume of, 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 and its complex version. The main tool used in the proofs is the conic version of the Crofton formula.

AB - The Gaussian polytope is the convex hull of n independent standard normally distributed points in. We derive explicit expressions for the probability that contains a fixed point as a function of the Euclidean norm of x, and the probability that contains the point, where is constant and X is a standard normal vector independent of. As a by-product, we also compute the expected number of k-faces and the expected volume of, 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 and its complex version. The main tool used in the proofs is the conic version of the Crofton formula.

KW - absorption probability

KW - average number of faces

KW - conic Crofton formula

KW - convex cone

KW - Convex hull

KW - error function

KW - Gaussian polytope

KW - Goodman-Pollack model

KW - random polytope

KW - regular simplex

KW - Schläfli's function

KW - solid angle

KW - spherical geometry

KW - Wendel's formula

UR - http://www.scopus.com/inward/record.url?scp=85088969600&partnerID=8YFLogxK

U2 - 10.1017/apr.2020.7

DO - 10.1017/apr.2020.7

M3 - Article

AN - SCOPUS:85088969600

VL - 52

SP - 588

EP - 616

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 2

ER -