Optimal Transport: Convexity and Inequalities

This lecture covers the concept of optimal transport, focusing on convexity properties and inequalities. Starting with the definition of optimal coupling, the instructor explains the Wassenstein geodesic and its relation to convex functions. The lecture delves into the application of Brunn-Minkowski inequality, demonstrating how it applies to compact sets in Rd. Through a series of proofs and examples, the lecture showcases the importance of convexity in optimal transport problems and how it leads to various inequalities.