## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 1770-1793.

### Uniform behaviors of random polytopes under the Hausdorff metric

#### 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.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 1770-1793.

**Dates**

Received: September 2017

Revised: March 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326427

**Digital Object Identifier**

doi:10.3150/18-BEJ1035

**Mathematical Reviews number (MathSciNet)**

MR3961230

**Zentralblatt MATH identifier**

07066239

**Keywords**

computational geometry convex bodies convex hull deviation inequality Hausdorff metric high dimension minimax estimation random polytope

#### Citation

Brunel, Victor-Emmanuel. Uniform behaviors of random polytopes under the Hausdorff metric. Bernoulli 25 (2019), no. 3, 1770--1793. doi:10.3150/18-BEJ1035. https://projecteuclid.org/euclid.bj/1560326427