Student Probability Seminar

What is the midpoint between two probability densities \mu and \nu, say, on the real line? Of course, the usual (\mu + \nu)/2 is a possibility. Here is another one: design a "transportation plan" between \mu and \nu, and stop halfway. Functionals that are convex for this interpolation are said to be "displacement convex".

This notion was introduced by McCann, and applied to prove uniqueness of minimisers for some functionals which are not convex in the usual sense. I'll present the basics of optimal transport, define displacement convexity and give an application.