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arXiv · 1109.5287

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

Sergey Bobkov·Mokshay Madiman
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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 Plünnecke-Ruzsa inequalities from additive combinatorics.

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Sergey Bobkov, Mokshay Madiman. 2011-09-24. Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures. https://doi.org/10.1016/j.jfa.2012.01.011

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