Random affine simplexes

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 $k\in\{1,\dots,d\}$ consider random vectors $X_0,\dots, X_{k}\in\mathbb R^d$ with an arbitrary spherically symmetric joint density function. Let $A$ be any non-singular $d\times d$ matrix. We show that the $k$-dimensional volume of the convex hull of affinely transformed $X_{i}$'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 $\mathcal{E}:=\{\mathbf{x}\in\mathbb R^d:{\mathbf{x}^\top (A^\top A)^{-1}\mathbf{x}}\leq 1\}$ is an ellipsoid, $P_\xi$ denotes the orthogonal projection to a random uniformly chosen $k$-dimensional linear subspace $\xi$ independent of $X_0,\dots, X_{k}$, and $\kappa_k$ is the volume of the unit $k$-dimensional ball. We express $|P_\xi\mathcal{E}|$ in terms of Gaussian random matrices. The important special case $k=1$ corresponds to the distance between two random points: \[ |AX_0-AX_1|\stackrel{d}{=}\sqrt{\frac{\lambda_1^2N_1^2+\dots+\lambda_d^2N_d^2}{N_1^2+\dots+N_d^2}}\cdot|X_0-X_1|, \] where $N_1,\dots,N_d$ are i.i.d. standard Gaussian variables independent of $X_0,X_1$ and $\lambda_1,\dots,\lambda_d$ are the singular values of $A$. As an application, we derive the following integral geometry formula for ellipsoids: \[ \frac{\kappa_{d}^{k+1}}{\kappa_k^{d+1}}\,\frac{\kappa_{k(d+p)+k}}{\kappa_{k(d+p)+d}}\,\int\limits_{A_{d,k}}|\mathcal{E}\cap E|^{p+d+1}\,\mu_{d,k}(dE)=|\mathcal{E}|^{k+1}\,\int\limits_{G_{d,k}}|P_L\mathcal{E}|^p\,\nu_{d,k}(dL), \] where $p> -d+k-1$ and $A_{d,k}$ and $G_{d,k}$ are the affine and the linear Grassmannians equipped with their respective Haar measures. The case $p=0$ reduces to an affine version of the integral formula of Furstenberg and Tzkoni.