Past lectures

Abstract

In this course we discuss various versions of the isoperimetric Inequality, namely, the Euclidean, the Spherical, the Hyperbolic, the Gaussian, and the Anisotropic versions.

Along the way, we present the classical Brunn-​Minkowski inequality and its functional version, namely, the Prekopa-​Leindler inequality. Since extremal bodies in various inequalities tend to be convex, fundamental properties of convex sets will be discussed.

The next topic, considered both from the discrete (polytopes) and the analytic (bodies with smooth boundary) point of view, is that the volume of non-​negative linear combinations of convex bodies leads to a polynomial expression. We shall relate this to analogues properties of the determinant and of the intersection numbers of algebraic hypersurfaces.

Finally, we discuss versions of the Minkowski problem (corresponding to a Monge-​Ampère equation on the sphere), and the relation of the uniqueness of their solution to Brunn-​Minkowski type inequalities.

A recent, dynamically expanding theory focuses on Lp and similar versions of the Minkowski problem, and leads to fundamental open questions linking Brunn-​Minkowski Theory, Probability Theory, and Monge-​Ampère equations.

During the course, a variety of methods will be introduced. For example, underlying ideas of the various proofs of the Brunn-​Minkowski inequality include combinatorics, spectral theory, uniqueness of the solution of a Monge-​Ampère equation, and optimal transportation.