# Absorption probabilities for Gaussian polytopes and regular spherical simplices

## TL;DR

This paper derives explicit formulas for the absorption probabilities of Gaussian polytopes, including their volume and face counts, using spherical simplex volumes and Crofton formulas, extending classical results in stochastic geometry.

## Contribution

It provides new explicit expressions for Gaussian polytope absorption probabilities and related geometric measures, connecting them with spherical simplex volumes and complex normal distribution functions.

## Key findings

- Explicit formulas for point containment probabilities in Gaussian polytopes.
- Expected number of faces and volume of Gaussian polytopes computed.
- Formulas expressed via standard normal distribution and spherical simplex volumes.

## 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.