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.
"Cone points of Brownian motion in arbitrary dimension." Ann. Probab. 47 (5) 3143 - 3169, September 2019. https://doi.org/10.1214/19-AOP1335