  This paper deals with sufficient conditions on the distribution of the random
variable $H$, in the model $X =\Pi_C(H)$, for the convex hull $\widehat C_N$ of
$N$ independent copies of $X$ to be a consistent estimator of the convex body
$C$ with a rate of convergence. The convergence of $\widehat C_N$ is
established for the Hausdorff distance under a uniform condition on the
distribution of $H$, but also in a pointwise sense under a less demanding
condition. Some of these convergence results on $\widehat C_N$ are applied to
the estimation of the time-dependent constraint set involved in a discrete-time
Skorokhod problem.
