  We investigate the properties of convex functions in the plane that satisfy a
local inequality which generalizes the notion of sub-solution of Monge-Ampere
equation for a Monge-Kantorovich problem with quadratic cost between
non-absolutely continuous measures. For each measure, we introduce a discrete
scale so that the measure behaves as an absolutely continuous measure up to
that scale. Our main theorem then proves that such convex functions cannot
exhibit any flat part at a scale larger than the corresponding discrete scales
on the measures. This, in turn, implies a $C^1$ regularity result up to the
discrete scale for the Legendre transform. Our result applies in particular to
any Kantorovich potential associated to an optimal transportation problem
between two measures that are (possibly only locally) sums of uniformly
distributed Dirac masses. The proof relies on novel explicit estimates directly
based on the optimal transportation problem, instead of the Monge-Ampere
equation.
