  We prove sharp $L^3$ bounds for a variable coefficient generalization of the
Wolff circular maximal function $M^\delta f$. For each fixed radius $r$,
$M^\delta f(r)$ is the maximal average of $f$ over the $\delta$--neighborhood
of a circle of radius $r$ and arbitrary center. In this paper, we consider
maximal averages over families of curves satisfying the cinematic curvature
condition, which was first introduced by Sogge to generalize the Bourgain
circular maximal function. Our proof manages to avoid a key technical lemma in
Wolff's original argument, and thus our proof also yields a shorter proof of
the boundedness of the (conventional) Wolff circular maximal function. At the
heart of the proof is an induction argument that employs an efficient
partitioning of $\mathbb{R}^3$ into cells using the discrete polynomial ham
sandwich theorem.
