Displacement convexity

Summary

A functional on probability measures is displacement convex if it is convex along the paths of optimal transport, rather than along straight-line mixtures of measures. Robert McCann introduced the notion in 1997, and used it to prove uniqueness of minimizers for energies that are not convex in the usual sense. In the language of the Wasserstein distance, displacement convexity is convexity along geodesics. Key examples are potential energies with a convex potential, interaction energies with a convex kernel, and the entropy $\int \rho \log \rho$.