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 $\mathbbR^n$, 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.