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