Title: 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 $d\geq 3$, 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
$\langle \mathcal{F}_T^{(d)}\rangle \sim 2\left[\ln \left( T/\Delta
t\right)\right]^{d-1}$, where $T$ is the total duration of the motion and
$\Delta t$ is the minimum time lapse separating points that define a facet.