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

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.