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. The specialization of this inequality to log-concave measures may be seen as a version of Milman's reverse Brunn-Minkowski inequality. N2 - We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. National Science Foundation CAREER grant DMS-1056996. was supported by a Junior Faculty Fellowship from Yale University and the U.S. National Science Foundation grant DMS-1106530, and M.M. T1 - Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures
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