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It is well known that nonlinear diffusion equations can be interpreted as agradient flow in the space of probability measures equipped with the EuclideanWasserstein distance. Under suitable convexity conditions on the nonlinearity,due to R. J. McCann, the associated entropy is geodesically convex, whichimplies a contraction type property between all solutions with respect to thisdistance. In this note, we give a simple straightforward proof of theequivalence between this contraction type property and this convexitycondition, without even resorting to the entropy and the gradient flowstructure.

Nonlinear diffusion: Geodesic Convexity is equivalent to Wasserstein Contraction