  We consider the convex hull of a finite sample of i.i.d. points uniformly
distributed in a convex body in $\R^d$, $d\geq 2$. We prove an exponential
deviation inequality, which leads to rate optimal upper bounds on all the
moments of the missing volume of the convex hull, uniformly over all convex
bodies of $\R^d$, with no restriction on their volume, location in the space
and smoothness of the boundary.
