Random affine simplexes
Friedrich G\"otze, Anna Gusakova, and Dmitry Zaporozhets

TL;DR
This paper investigates the distribution of volumes of convex hulls of randomly transformed vectors and derives geometric formulas involving ellipsoids, Gaussian matrices, and affine transformations, extending classical integral geometry results.
Contribution
It provides a novel distributional identity for the volume of affinely transformed simplexes and derives new integral geometry formulas for ellipsoids involving random projections.
Findings
Distributional formula for convex hull volumes under affine transformations
Representation of projected ellipsoid volumes via Gaussian matrices
New integral geometry formula for ellipsoids and affine Grassmannians
Abstract
For a fixed consider random vectors with an arbitrary spherically symmetric joint density function. Let be any non-singular matrix. We show that the -dimensional volume of the convex hull of affinely transformed 's satisfies \[ |\mathrm{conv}(AX_0,\dots,AX_{k})|\stackrel{d}{=}\frac{|P_\xi\mathcal{E}|}{\kappa_k}\cdot|\mathrm{conv}(X_0,\dots,X_{k})|, \] where is an ellipsoid, denotes the orthogonal projection to a random uniformly chosen -dimensional linear subspace independent of , and is the volume of the unit -dimensional ball. We express in terms of Gaussian random matrices. The important special case corresponds to the distance between…
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