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

Details Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures

2011-09-24

arxiv.org/abs/1109.5287v1

Journal of Functional Analysis, Vol. 262, no. 7, pp. 3309-3339, April 2012

Mathematics

Functional Analysis

28 pages, revised version of a document submitted in October 2010

Scientific paper

10.1016/j.jfa.2012.01.011

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.

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