The optimal transport problem can be summarized as, given two probability densities, finding the cheapest way to transport one onto the other.
After introducing some basic knowledge about optimal transport among which existence of a solution, entropic regularization and dual formulation, we will present more recent work giving explicit stability
bounds. Quite interestingly, the main tool of the proof is Prékopa-Leindler inequality, a functional inequality one can actually prove using optimal transport, thus yielding a nice self improvement of the theory.