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 d-dimensional space 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.
The preprint on my homepage contains as an additional appendix a self-contained proof of the ε-net lemma from Komlós, Pach, and Woeginger [1992], which is used to show the existence of ε-nets.
Last update: October 12, 2020.