  We prove two-sided bounds on the expected values of several geometric
functionals of the convex hull of Brownian motion in $\mathbb{R}^n$ and their
inverse processes. This extends some recent results of McRedmond and Xu (2017),
Jovaleki\'{c} (2021), and Cygan, \v{S}ebek, and the first author (2023) from
the plane to higher dimensions. Our main result shows that the average time
required for the convex hull in $\mathbb{R}^n$ to attain unit volume is at most
$n\sqrt[n]{n!}$. Moreover, all of our bounds capture the correct order of
asymptotic growth or decay in the dimension $n$.
