  We investigate existence of dual optimizers in one-dimensional martingale
optimal transport problems. While [BNT16] established such existence for weak
(quasi-sure) duality, [BHP13] showed existence for the natural stronger
pointwise duality may fail even in regular cases. We establish that (pointwise)
dual maximizers exist when $y\mapsto c(x,y)$ is convex, or equivalent to a
convex function. It follows that when marginals are compactly supported, the
existence holds when the cost $c(x,y)$ is twice continuously differentiable in
$y$. Further, this may not be improved as we give examples with $c(x,\cdot)\in
C^{2-\epsilon}$, $\epsilon >0$, where dual attainment fails. Finally, when
measures are compactly supported, we show that dual optimizers are Lipschitz if
$c$ is Lipschitz.
