A relaxation approach to optimal transport with applications to the unbalanced case

In this talk, after briefly presenting the classical optimal transport problem, we will discuss a new interpretation of the optimal transport cost as the largest lower semicontinuous convex functional extending the cost between pairs of delta measures.
With this in mind, we will introduce a notion of optimal transport cost for (nonnegative) measures with possibly different masses and discuss its metric and topological properties.
This talk is based on a joint work with Giuseppe Savaré (Bocconi University).