  This paper deals with the existence of optimal transport maps for some
optimal transport problems with a convex but non strictly convex cost. We give
a decomposition strategy to address this issue. As part of our strategy, we
have to treat some transport problems, of independent interest, with a convex
constraint on the displacement. As an illustration of our strategy, we prove
existence of optimal transport maps in the case where the source measure is
absolutely continuous with respect to the Lebesgue measure and the
transportation cost is of the form h(||x-y||) with h strictly convex increasing
and ||. || an arbitrary norm in \R2.
