Tukey's halfaspace depth has attracted much interest in data analysis, because it is a natural way of measuring the notion of depth relative to a cloud of points or, more generally, to a probability measure. Given an i.i.d. sample, we investigate the concentration of upper level sets of the Tukey depth relative to that sample around their population version. We show that under some mild assumptions on the underlying probability measure, concentration occurs at a parametric rate and we deduce moment inequalities at that same rate. In a computational prospective, we study the concentration of a discretized version of the empirical upper level sets.

We estimate convex polytopes and general convex sets in R d , d ≥ 2 in the regression framework. We measure the risk of our estimators using a L 1 -type loss function and prove upper bounds on these risks. We show, in the case of convex polytopes, that these estimators achieve the minimax rate. For convex polytopes, this minimax rate is ln n n , which differs from the parametric rate for non-regular families by a logarithmic factor, and we show that this extra factor is essential. Using polytopal approximations we extend our results to general convex sets, and we achieve the minimax rate up to a logarithmic factor. In addition we provide an estimator that is adaptive with respect to the number of vertices of the unknown polytope, and we prove that this estimator is optimal in all classes of convex polytopes with a given number of vertices.

We estimate the support of a uniform density, when it is assumed to be a convex polytope or, more generally, a convex body in R d . In the polytopal case, we construct an estimator achieving a rate which does not depend on the dimension d, unlike the other estimators that have been proposed so far. For d ≥ 3, our estimator has a better risk than the previous ones, and it is nearly minimax, up to a logarithmic factor. We also propose an estimator which is adaptive with respect to the structure of the boundary of the unknown support.