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

TL;DR

This paper establishes a reverse entropy power inequality for convex measures, linking information theory and convex geometry, with implications for high-dimensional probability and additive combinatorics.

Contribution

It introduces a novel reverse entropy power inequality for convex measures, extending geometric inequalities to an information-theoretic context.

Findings

Derived a reverse inequality for convex measures.

Connected entropy of high-dimensional vectors with convex body volume.

Provided a continuous analogue of Plünnecke-Ruzsa inequalities.

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 Pl\"unnecke-Ruzsa inequalities from additive combinatorics.