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 BrunnMinkowski inequality.
3) Mixed volumes: basic properties and the AleksandrovFenchel inequality.
4) KlainSchneider 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 BrunnMinkowski Theory"(a book)
P. McMullen, "Valuations and Dissections"(a survey article, 1993).
Prerequisites: introduction to functional analysis on the level of
the HahnBanach 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.
