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Absorption probabilities for Gaussian polytopes and regular spherical simplices

Part of: Stochastic processes Geometric probability and stochastic geometry Polytopes and polyhedra General convexity

Published online by Cambridge University Press:  15 July 2020

Zakhar Kabluchko  and
Dmitry Zaporozhets
Zakhar Kabluchko*
Affiliation:
Westfälische Wilhelms-Universität Münster
Dmitry Zaporozhets*
Affiliation:
St. Petersburg Department of Steklov Mathematical Institute
*
*Postal address: Orléans–Ring 10, 48149 Münster, Germany. Email: zakhar.kabluchko@uni-muenster.de
**Postal address: Fontanka 27, 191011 St. Petersburg, Russia.
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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.

Keywords

Convex hullrandom polytopeGaussian polytopeGoodman–Pollack modelabsorption probabilityWendel’s formularegular simplexspherical geometrysolid angleconvex coneconic Crofton formulaaverage number of faceserror functionSchläfli’s function

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry 52B11: $n$-dimensional polytopes
Secondary: 60G70: Extreme value theory; extremal processes 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A23: Asymptotic theory of convex bodies
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 588 - 616
Copyright
© Applied Probability Trust 2020

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