  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
$d\geq 3$, 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.
