Given a functional inequality whose extremizers are known, the question of stability is as follows: If a function almost saturates the inequality, is it close to some extremizer? The most famous example is perhaps that of the isoperimetric inequality: The extremizers are the balls, and the question of stability comes down to showing that the isoperimetric deficit controls a certain distance from the ball. There are 4 methods for obtaining such results: the symmetrisation method, the transport mass approach, the selection principle and the ABP method. In this talk, I will present a recent work in which I introduce a fifth method, Stein's method. In particular, I will show how it proves a stability result for the first Steklov eigenvalue.

In addition, I will also present a stability result for the Brascamp-Lieb inequality, which is a functional inequality encoding certain weighted isoperimetric properties. This last result is based on joint work in progress with Bonnefont (IMB Bordeaux) and Joulin (IMT Toulouse).