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

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