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We investigate the properties of convex functions in the plane that satisfy alocal inequality which generalizes the notion of sub-solution of Monge-Ampereequation for a Monge-Kantorovich problem with quadratic cost betweennon-absolutely continuous measures. For each measure, we introduce a discretescale so that the measure behaves as an absolutely continuous measure up tothat scale. Our main theorem then proves that such convex functions cannotexhibit any flat part at a scale larger than the corresponding discrete scaleson the measures. This, in turn, implies a $C^1$ regularity result up to thediscrete scale for the Legendre transform. Our result applies in particular toany Kantorovich potential associated to an optimal transportation problembetween two measures that are (possibly only locally) sums of uniformlydistributed Dirac masses. The proof relies on novel explicit estimates directlybased on the optimal transportation problem, instead of the Monge-Ampereequation.

Local regularity result for an optimal transportation problem with rough measures in the plane