Title: Uniform deviation and moment inequalities for random polytopes with general densities in arbitrary convex bodies,
Abstract: We prove an exponential deviation inequality for the convex hull of a finite
sample of i.i.d. random points with a density supported on an arbitrary convex
body in $\R^d$, $d\geq 2$. When the density is uniform, our result yields rate
optimal upper bounds for all the moments of the missing volume of the convex
hull, uniformly over all convex bodies of $\R^d$: We make no restrictions on
their volume, location in the space or smoothness of their boundary. After
extending an identity due to Efron, we also prove upper bounds for the moments
of the number of vertices of the random polytope. Surprisingly, these bounds do
not depend on the underlying density and we prove that the growth rates that we
obtain are tight in a certain sense.