Output-sensitive Convex Hull in R^d

Let $d$ be fixed, let $S$ be a set of $n$ points in $\mathbb{R}^d$, and let the boundary complex of $\operatorname{conv}(S)$ have $f$ faces. Determine whether the convex hull can be constructed in $O(n\log f+f)$ time, or otherwise determine the tight worst-case output-sensitive complexity as a function of $n$, $f$, and $d$.