Functional inequalities

*The speaker is also affiliated with Max Planck Institute of Mathematics

The hyperplane problem asks whether there exists an absolute constant such that every symmetric convex body of volume one in every dimension has a central hyperplane section with area greater than this constant. We consider a generalization of this problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional

We discuss entropy inequalities for log concave functions. Among them
are reverse log Sobolev inequalities. This leads naturally to a concept
of relative entropy and Renyi entropy for such functions.

Connections are given to the theory of convex bodies.

In this lecture we will review some classic and more recent results on the class of log-concave functions, focusing on the analogies with the theory of convex bodies. We will be particularly interested in functional inequalities. The main example will be the Prékopa-Leindler inequality, that we will present along with its infinitesimal version; but we will also see other examples like the functional versions of the Blashke-Santaló inequality and of Rogers-Shephard inequality.

I will present the results of a recent work in collaboration with Daria Ghilli (Università di Padova). We give stability results for the Borell-Brascamp-Lieb inequality for compactly supported power concave functions and apply these results to strengthen the Brunn-Minkowski and the Urysohn inequalities for the torsional rigidity of convex sets.