This is an announcement for the paper "Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures" by Sergey Bobkov and Mokshay Madiman.
Abstract: 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.
Archive classification: math.FA math.PR
Remarks: 28 pages, revised version of a document submitted in October 2010
Submitted from: mokshay.madiman@yale.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1109.5287
or