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

Sergey Bobkov, Mokshay Madiman

Research output: Contribution to journalArticle

37 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)3309-3339
Number of pages31
JournalJournal of Functional Analysis
Volume262
Issue number7
DOIs
StatePublished - Apr 1 2012

Keywords

  • Brunn-Minkowski inequality
  • Convex measure
  • Entropy
  • Log-concave

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