  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. 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 Pl\"unnecke-Ruzsa inequalities from additive
combinatorics.
