Adaptive estimation of convex and polytopal density support

We estimate the support of a uniform density, when it is assumed to be a convex polytope or, more generally, a convex body in Rd. 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.