Forward and reverse entropy power inequalities

Abstract: The entropy power inequality plays a fundamental role in information theory and probability (for instance, it sheds light on both the central limit theorem and the Gaussian logarithmic Sobolev inequality). In the first part of the lectures, we survey various recent developments on forward and reverse entropy power inequalities not just for the Shannon-Boltzmann entropy but also more generally for Rényi entropy. In the process, we discuss connections of these inequalities to both Convex Geometry and Additive Combinatorics. In the second part of these lectures, we discuss specifically the case of the Rényi entropy of order infinity. By combining an extreme point characterization in a space of measures with new geometric inequalities, we obtain lower bounds for the Rényi entropy of the sum in this case that improve upon and unify results of Rogozin, Bobkov-Chistyakov, Rudelson-Vershynin, and Livshyts-Paouris-Pivovarov. If time permits, we will also discuss reverse inequalities in this setting.