Lampros Gavalakis (University of Cambridge) – Information-theoretic analogues of Bonnesen's and Bergström's inequalities

The entropy of a continuous random variable behaves, in some ways, similarly to the logarithm of the volume of a set. In particular, the Entropy Power Inequality (EPI) is widely considered as an information-theoretic analogue of the Brunn-Minkowski inequality. In fact, a common proof of the two inequalities exists via a sharp form of Young's inequality.

After briefly reviewing this connection between convex geometry and information theory, we will present a new inequality for entropy, which improves the EPI under assumptions on the marginals, in the same way that Bonnesen's inequality improves the Brunn-Minkowski inequality under assumptions on the volume along some projection. We will characterize the equality case in the latter inequality. Furthermore, we will show how this inequality follows from a more general entropy inequality, which reduces to Bergström's inequality for determinants in the Gaussian case. We will also discuss a related inequality for the Fisher information.

This talk is based on joint work with Matthieu Fradelizi and Martin Rapaport.