Gaussian Convex Bodies: a Nonasymptotic Approach

Abstract

We study linear images of a symmetric convex body C ⊆ ℝ^N under an n × N Gaussian random matrix G, where N ≥ 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 × 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 ⊆ ℝ^N, with N → ∞, we obtain new asymptotic results and take first steps to compare with the asymptotic theory.