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 ${\mathbb{R}}^d$, $d\ge 2$. 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 ${\mathbb{R}}^d$, with no restriction on their volume, location in the space and smoothness of the boundary.

References in corpus (1)

• Adaptive estimation of convex and polytopal support

Cited by in corpus (2)

• Estimation of convex supports from noisy measurements
• The wrapping hull and a unified framework for volume estimation