Random affine simplexes

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 \[ |\mathrm{conv}(AX_0,\dots,AX_{k})|\stackrel{d}{=}\frac{|P_ξ\mathcal{E}|}{κ_k}\cdot|\mathrm{conv}(X_0,\dots,X_{k})|, \] where $\mathcal{E}:=\{\mathbf{x}\in {\mathbb{R}}^d:{\mathbf{x}}^{\top }(A^{\top }A{)}^{-1}\mathbf{x}\le 1\}$ 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: \[ |AX_0-AX_1|\stackrel{d}{=}\sqrt{\frac{λ_1^2N_1^2+\dots+λ_d^2N_d^2}{N_1^2+\dots+N_d^2}}\cdot|X_0-X_1|, \] 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: \[ \frac{κ_{d}^{k+1}}{κ_k^{d+1}}\,\frac{κ_{k(d+p)+k}}{κ_{k(d+p)+d}}\,\int\limits_{A_{d,k}}|\mathcal{E}\cap E|^{p+d+1}\,μ_{d,k}(dE)=|\mathcal{E}|^{k+1}\,\int\limits_{G_{d,k}}|P_L\mathcal{E}|^p\,ν_{d,k}(dL), \] 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.

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