Grassmann angles and absorption probabilities of Gaussian convex hulls

TL;DR

This paper establishes a connection between the probability of the origin being inside the convex hull of a Gaussian image of a set and the Grassmann angles of its conic hull, providing new insights into Gaussian convex hulls.

Contribution

It introduces a novel relationship between absorption probabilities of Gaussian convex hulls and Grassmann angles, and analyzes expected Grassmann angles of Gaussian images.

Findings

Probability that convex hull contains the origin equals the $k$-th Grassmann angle of the conic hull.

Expected Grassmann angles of Gaussian images match those of the original cone.

Expected angle sums of Gaussian simplices match those of regular simplices.

Abstract

Let $M$ be an arbitrary subset in $\mathbb R^n$ with a conic (or positive) hull $C$. Consider its Gaussian image $AM$, where $A$ is a $k\times n$-matrix whose entries are independent standard Gaussian random variables. We show that the probability that the convex hull of $AM$ contains the origin in its interior coincides with the $k$-th Grassmann angle of $C$. Also, we prove that the expected Grassmann angles of $AC$ coincide with the corresponding Grassmann angles of $C$. Using the latter result, we show that the expected sum of $j$-th Grassmann angles at $\ell$-dimensional faces of a Gaussian simplex equals the analogous angle-sum for the regular simplex of the same dimension.