Poincaré Inequalities and Moment Maps

We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in ℝ n . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of ℓ p -spaces in ℝ n for 0 < p < 1 .