Cone points of Brownian motion in arbitrary dimension

TL;DR

This paper proves that the convex hull of an n-dimensional Brownian motion path is smooth and that Brownian motion almost surely has no cone points for cones with nontrivial dual cones.

Contribution

It establishes the smoothness of the convex hull of Brownian motion in arbitrary dimensions and characterizes the absence of cone points.

Findings

Convex hull of Brownian motion is smooth in any dimension.

Brownian motion almost surely has no cone points for cones with nontrivial duals.

Results hold for Brownian motion up to time 1.

Abstract

We show that the convex hull of the path of Brownian motion in $n$-dimensions, up to time $1$, is a smooth set. As a consequence, we conclude that a Brownian motion in any dimension almost surely has no cone points for any cone whose dual cone is nontrivial.