Random polytopes and the wet part for arbitrary probability distributions

Abstract

We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in ${\mathbf{R}}^d$ relates to the wet part of that measure. This extends classical results for the uniform distribution from a convex set [Bárány and Larman 1988]. The lower bound of Bárány and Larman continues to hold in the general setting, but the upper bound must be relaxed by a factor of $\log n$. We show by an example that this is tight.

13 pages, 1 figure