Isoperimetric Inequality: Optimal Transport

This lecture focuses on the isoperimetric inequality and its surprising application in optimal transport. The instructor explains how the Brignier theorem is used to prove the inequality, showing that the perimeter of a set is greater or equal to the perimeter of a ball with the same volume. The lecture delves into the properties of the optimal transport map, the convexity of functions, and the uniqueness of optimizers. The instructor also discusses quality cases, where sets other than a ball minimize the perimeter at a given area. The lecture concludes with a discussion on the deficit of a set and a refined theorem by Alessio Figalli. The concepts are illustrated with examples and proofs, providing insights into the geometric implications of the isoperimetric inequality.