# Facets on the convex hull of $d$-dimensional Brownian and L\'evy motion

## TL;DR

This paper derives an exact, universal formula for the average number of facets on the convex hull of $d$-dimensional stationary Markov processes, with explicit results for Brownian motion relevant to various scientific fields.

## Contribution

It provides a universal, exact formula for the average number of facets on the convex hull of $d$-dimensional Markov processes, including Brownian motion, independent of increment distribution.

## Key findings

- Exact formula for average number of facets in $d ext{-}dim$ processes.
- Universality class independent of increment distribution.
- Asymptotic behavior scales as $( ext{ln}(T/ riangle t))^{d-1}$.

## Abstract

For stationary, homogeneous Markov processes (viz., L\'{e}vy 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.