Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes

Abstract

We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave probability measures on ${\mathbb{R}}^d$ with full support. Our results extend the corresponding Gaussian lower bounds of Bárány and Vu and of Bárány and Thäle to this broader class of distributions. Our proof builds on the geometric construction introduced by Bárány and Vu, for which we develop an alternative treatment based on barycentric coordinates. We combine it with local variance estimates for both intrinsic volumes and the number of faces.