Bernoulli

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

Uniform behaviors of random polytopes under the Hausdorff metric

Victor-Emmanuel Brunel

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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


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