Smoothness of the Convex Hull of Planar Brownian Motion

In this article we prove that for each $t > 0$, almost surely $\partial C(t)$, the boundary of the convex hull of two dimensional Brownian motion up to time $t$, is a $C^1$ curve in the plane. We also prove that if $\eta$ is a modulus of continuity such that $x\eta(x)$ is convex and $\int^1_0\eta(x) dx/x < \infty$ then for each $t > 0$, almost surely $\partial C(t)$ is not a $C^{1, \eta}$ curve in the plane.