Volumetric Properties of the Convex Hull of an n-dimensional Brownian Motion

TL;DR

This paper derives formulas for the expected volume and surface area of the convex hull of an n-dimensional Brownian motion, analyzes approximation by discrete steps, and explores the scale invariance of its facets.

Contribution

It provides new formulas for geometric properties of Brownian convex hulls and investigates their approximation and facet distribution behaviors.

Findings

Expected volume and surface area formulas derived

Number of steps for accurate discrete approximation scales as n^3

Facet distribution exhibits scale invariance as size increases

Abstract

Let K be the convex hull of the path of a standard brownian motion B(t) in R^n, taken at time 0 < t < 1. We derive formulas for the expected volume and surface area of K. Moreover, we show that in order to approximate K by a discrete version of K, namely by the convex hull of a random walk attained by taking B(t_n) at discrete (random) times, the number of steps that one should take in order for the volume of the difference to be relatively small is of order n^3. Next, we show that the distribution of facets of K is in some sense scale invariant: for any given family of simplices (satisfying some compactness condition), one expects to find in this family a constant number of facets of tK as t approaches infinity. Finally, we discuss some possible extensions of our methods and suggest some further research.