Geodesic
Grassmann angles and absorption probabilities of Gaussian convex hulls
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 30, Tome 501 (2021), pp. 126-148

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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. 
@article{ZNSL_2021_501_a8,
     author = {F. G\"otze and Z. Kabluchko and D. Zaporozhets},
     title = {Grassmann angles and absorption probabilities of {Gaussian} convex hulls},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {126--148},
     publisher = {mathdoc},
     volume = {501},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_501_a8/}
}
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F. Götze; Z. Kabluchko; D. Zaporozhets. Grassmann angles and absorption probabilities of Gaussian convex hulls. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 30, Tome 501 (2021), pp. 126-148. http://geodesic.mathdoc.fr/item/ZNSL_2021_501_a8/