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 language | English (US) |
|---|---|
| Pages (from-to) | 3309-3339 |
| Number of pages | 31 |
| Journal | Journal of Functional Analysis |
| Volume | 262 |
| Issue number | 7 |
| DOIs | |
| State | Published - Apr 1 2012 |
Bibliographical note
Funding Information:✩ S.B. was supported in part by U.S. National Science Foundation grant DMS-1106530, and M.M. was supported by a Junior Faculty Fellowship from Yale University and the U.S. National Science Foundation CAREER grant DMS-1056996. * Corresponding author. Fax: +1 203 432 0633. E-mail addresses: [email protected] (S. Bobkov), [email protected] (M. Madiman).
Keywords
- Brunn-Minkowski inequality
- Convex measure
- Entropy
- Log-concave