Random polytopes and estimation of convex bodies

In this talk we discuss properties of random polytopes. In particular, we study the convex hull of i.i.d. random points, whose law is supported on a convex body. We propose deviation and moment inequalities for this random polytope, and then discuss its optimality, when it is seen as an estimator of the support of the probability measure, which may be unknown.
We also define a notion of multidimensional quantile sets for probability measures in a Euclidean space. These are convex sets, which are related to the notions of floating bodies, or the Tukey depth. When i.i.d. random points are available, these multidimensional quantile sets can be estimated by their empirical versions, and we again propose deviation and moment inequalities for the empirical quantile sets.