Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures
arXiv:1109.5287 · doi:10.1016/j.jfa.2012.01.011
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 -ellipsoids and on certain information-theoretic inequalities. As a by-product, we also give a continuous analogue of some Plünnecke-Ruzsa inequalities from additive combinatorics.
28 pages, revised version of a document submitted in October 2010
References in corpus (3)
Cited by in corpus (25)
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