Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

We develop a reverse entropy power inequality for convex measures, which may be seen as an affinegeometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman’s reverse Brunn–Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman’s deep technology of M-ellipsoids and on certain information-theoretic inequalities. As a by-product, we also give a continuous analogue of some Plünnecke–Ruzsa inequalities from additive combinatorics. © 2012 Elsevier Inc. All rights reserved.