Random Convex Hulls and Extreme Value Statistics

TL;DR

This paper develops a unifying method to compute the mean perimeter and area of convex hulls formed by random points, including correlated points like Brownian motions, linking convex geometry with extreme value statistics.

Contribution

It introduces a novel approach using support functions and Cauchy's formulae to exactly calculate convex hull properties for both independent and correlated random points.

Findings

Exact formulas for mean perimeter and area of convex hulls for independent points.

Extension of results to correlated points, including Brownian trajectories.

Demonstrates a link between convex hull statistics and extreme value theory.

Abstract

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].