New Connections Between the Entropy Power Inequality and Geometric Inequalities

Abstract

The entropy power inequality (EPI) has a fundamental role in Information Theory, and has deep connections with famous geometric inequalities. In particular, it is often compared to the Brunn-Minkowski inequality in convex geometry. In this article, we further strengthen the relationships between the EPI and geometric inequalities. Specifically, we establish an equivalence between a strong form of reverse EPI and the hyperplane conjecture, which is a long-standing conjecture in high-dimensional convex geometry. We also provide a simple proof of the hyperplane conjecture for a certain class of distributions, as a straightforward consequence of the EPI.

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