Gaussian convex bodies: a non-asymptotic approach

We study linear images of a symmetric convex body $C\subseteq\mathbb R^N$ under an $n\times N$ Gaussian random matrix $G$, where $N\ge n$. Special cases include common models of Gaussian random polytopes and zonotopes. We focus on the intrinsic volumes of $GC$ and study the expectation, variance, small and large deviations from the mean, small ball probabilities, and higher moments. We discuss how the geometry of $C$, quantified through several different global parameters, affects such concentration properties. When $n=1$, $G$ is simply a $1\times N$ row vector and our analysis reduces to Gaussian concentration for norms. For matrices of higher rank and for natural families of convex bodies $C_N\subseteq\mathbb R^N$, with $N\to\infty$, we obtain new asymptotic results and take first steps to compare with the asymptotic theory.