Dimensional behaviour of entropy and information

TL;DR

This paper explores the properties of entropy and information in convex geometry, introducing new inequalities and comparisons for log-concave measures, and offering insights into longstanding conjectures.

Contribution

It presents novel entropy-based inequalities and comparison results for log-concave measures, including a new equipartition property and a reverse entropy power inequality.

Findings

New equipartition property for log-concave measures

Gaussian comparison results for log-concave measures

A reverse entropy power inequality analogous to Milman's inequality

Abstract

We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman's reverse Brunn-Minkowski inequality.