Topics in Convexity: Theory of Valuations.

Here is a brief overview of the course "Topics in Convexity: theory of valuations.". The central notion discussed in the course is the notion of valuaiton on convex sets. A valuation is an additive functional on the family of convex compact subsets of a finite dimensional real vector space. The basic examples of valuations are Lebesgue measure and the Euler characteristic. More examples include mixed volumes (which will be discussed in the course in some detail).

Syllabus of the course:

1) Valuations on polytopes. McMullen's polytope algebra.

2) The Brunn-Minkowski inequality.

3) Mixed volumes: basic properties and the Aleksandrov-Fenchel inequality.

4) Klain-Schneider characterization of simple translation invariant continuous valuations.

5) Hadwiger's characterization of isometry invariant continuous valuations.

6) Irreducibility theorem and McMullen's conjecture on translation invariant continuous valuations.

7) The multiplicative structure on valuations.

8)Valuations invariant under a group.

9) Applications to integral geometry (if time permits).

For a part of the course the following literature will be useful:

R. Schneider, "Convex Bodies:the Brunn-Minkowski Theory"(a book)

P. McMullen, "Valuations and Dissections"(a survey article, 1993).

Prerequisites: introduction to functional analysis on the level of the Hahn-Banach theorem, the Banach inverse theorem, basics of the theory of generalized functions (distributions). Some familiarity with manifolds, differential forms, and vector bundles would be useful, but not strictly necessary.