Facets on the convex hull of d-dimensional Brownian and Lévy motion.

Abstract

For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension d≥3, we establish an exact formula for the average number of (d-1)-dimensional facets that can be defined by d points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when d=3, a case which is of particular interest for applications in biophysics, chemistry, and polymer science. We also show that the asymptotical average number of facets behaves as 〈F_{T}^{(d)}〉∼2[ln(T/Δt)]^{d-1}, where T is the total duration of the motion and Δt is the minimum time lapse separating points that define a facet.