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

A strategy for non-strictly convex transport costs and the example of ||x-y||p in R2