A universal deviation inequality for random polytopes

Abstract

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.