Uniform deviation and moment inequalities for random polytopes with general densities in arbitrary convex bodies

TL;DR

This paper establishes exponential deviation inequalities and moment bounds for the convex hull of i.i.d. random points with general densities in convex bodies, providing optimal rates and tight bounds independent of the density.

Contribution

It introduces new deviation inequalities and moment bounds for random polytopes with general densities, extending existing results to arbitrary convex bodies without smoothness restrictions.

Findings

Optimal upper bounds for moments of missing volume

Density-independent bounds for the number of vertices

Tight growth rates for these bounds

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.