Deviation inequalities for random polytopes 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 $\mathbb{R}^{d}$, $d\geq 2$. 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 $\mathbb{R}^{d}$: We make no restrictions on their volume, location in the space or smoothness of their boundary. For general densities, the only restriction we make is that the density is bounded from above, even though we believe this restriction is not necessary. However, the density can have any decay to zero near the boundary of its support. 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. Our results are non asymptotic and hold uniformly over all convex bodies.