Forward and Reverse Entropy Power Inequalities in Convex Geometry

TL;DR

This paper surveys recent advances in forward and reverse entropy power inequalities, exploring their connections to convex geometry and extending results beyond Shannon entropy to Rényi entropy.

Contribution

It provides a comprehensive overview of new developments linking entropy power inequalities with convex geometric inequalities, including both forward and reverse forms for various entropies.

Findings

Extended entropy power inequalities to Rényi entropy.

Explored connections between functional and probabilistic inequalities.

Reviewed recent progress in convex geometric interpretations of entropy inequalities.

Abstract

The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent developments on forward and reverse entropy power inequalities not just for the Shannon-Boltzmann entropy but also more generally for R\'enyi entropy. In the process, we discuss connections between the so-called functional (or integral) and probabilistic (or entropic) analogues of some classical inequalities in geometric functional analysis