In 2007, Bouchitté, Jimenez and Rajesh proved a “worst-case” estimate in optimal transport: under suitable assumptions, one can bound the largest distance that mass travels in the optimal transport map using the average quadratic transport cost. This type of estimate became a standard tool in analysis and PDE. In 2016, Rajala and Jylha revisited the theory in a more systematic way, clarifying the underlying mechanism, and identifying sharp structural conditions under which an average transport cost can control a worst-case transport cost.

In this talk I will first explain the idea behind these reverse estimates. I will then present the main results of a recent work in which extend this viewpoint beyond the increasing costs of the distance and beyond the classical two-marginal setting. In particular, the paper develops reverse inequalities for more general costs and for multi-marginal optimal transport, showing how worst-case transport energies can be controlled in terms of integral energies together with explicit concentration information on the common marginal.