On uniform performances of random polytopes and their functionals in convex bodies

In a previous paper we considered the random polytope defined as the convex hull of $n$ i.i.d. random points uniformly distributed in a convex body and we focused on the Nikodym distance between that convex body and the random polytope. We proved a uniform deviation inequality, yielding tight moment inequalities in a minimax sense. In this paper, we use a different technique in order to prove similar types of inequalities for the Hausdorff distance between the random polytope and the convex body. We no longer restrict ourselves to the case of uniform distributions. However, our results allow to recover and improve some known inequalities for the convex hull of uniform points in a convex body or on its boundary. We also prove moment inequalities for some class of functionals of the random polytope, including mean width as a special case.