  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.
