Uniform Behaviors of Random Polytopes under the Hausdorff Metric

TL;DR

This paper investigates the Hausdorff distance between random polytopes and their support convex bodies, revealing that smooth boundaries lead to better statistical accuracy than polytopal supports, with optimal rates and practical approximation methods.

Contribution

It provides new insights into the behavior of random polytopes under the Hausdorff metric, including rate optimality, extensions to rescaled metrics, and computationally efficient approximations.

Findings

Random polytopes have better Hausdorff accuracy with smooth boundaries.

Rates of convergence are proven to be minimax optimal.

High-dimensional polytopes can be approximated efficiently without losing accuracy.

Abstract

We study the Hausdorff distance between a random polytope, defined as the convex hull of i.i.d. random points, and the convex hull of the support of their distribution. As particular examples, we consider uniform distributions on convex bodies, densities that decay at a certain rate when approaching the boundary of a convex body, projections of uniform distributions on higher dimensional convex bodies and uniform distributions on the boundary of convex bodies. We essentially distinguish two types of convex bodies: those with a smooth boundary and polytopes. In the case of uniform distributions, we prove that, in some sense, the random polytope achieves its best statistical accuracy under the Hausdorff metric when the support has a smooth boundary and its worst statistical accuracy when the support is a polytope. This is somewhat surprising, since the exact opposite is true under the Nikodym metric. We prove rate optimality of most our results in a minimax sense. In the case of uniform distributions, we extend our results to a rescaled version of the Hausdorff metric. We also tackle the estimation of functionals of the support of a distribution such as its mean width and its diameter. Finally, we show that high dimensional random polytopes can be approximated with simple polyhedral representations that significantly decrease their computational complexity without affecting their statistical accuracy.