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

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 Milmanʼs reverse Brunn–Minkowski inequali...

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Bibliographic Details
Published in:Journal of functional analysis 2012-04, Vol.262 (7), p.3309-3339
Main Authors: Bobkov, Sergey, Madiman, Mokshay
Format: Article
Language:English
Subjects:
Brunn–Minkowski inequality
Convex measure
Entropy
Log-concave
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Summary: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 Milmanʼs 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 Milmanʼs 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.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2012.01.011