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(a)yale.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1109.5287
or
http://arXiv.org/abs/1109.5287
