Poincaré inequalities and moment maps (Q355210)

The paper under review contains a thorough discussion of a method for obtaining bounds on the variance of a function in terms of an integral of a quadratic form involving the gradient of the function. Its applicability is not restricted to convex sets. The main idea is to take a detour in a space of dimension two or three where the existence of additional symmetries allows an application of a dual version of Bochner's formula. The success is guaranteed for suitably restricted functions. In spite of many technical details, the paper is good to read because the author takes the time to explain the novelties and the delicate points in the approach.