  This paper concerns the regularity and geometry of the free boundary in the
optimal partial transport problem for general cost functions. More
specifically, we prove that a $C^1$ cost implies a locally Lipschitz free
boundary. As an application, we address a problem discussed by Caffarelli and
McCann \cite{CM} regarding cost functions satisfying the Ma-Trudinger-Wang
condition (A3): if the non-negative source density is in some
$L^p(\mathbb{R}^n)$ space for $p \in (\frac{n+1}{2},\infty]$ and the positive
target density is bounded away from zero, then the free boundary is a
semiconvex $C_{loc}^{1,\alpha}$ hypersurface. Furthermore, we show that a
locally Lipschitz cost implies a rectifiable free boundary and initiate a
corresponding regularity theory in the Riemannian setting.
