KASS seminar

Abstract: A convex body in $R^n$ is a compact convex set. A valuation is here a function on the space of convex bodies that behaves like a finitely additive measure,

$$
V(K\cup L) + V(K\cap L)= V(K)+V(L),
$$

provided that $K\cup L$ is convex. In spite of the simplicity of this definition, it turns out that translation invariant valuations have a lot of surprising properties (due to Alesker, Bernig, Fu, Kotrbatý Wannerer and others). On the one hand side the space of valuations resembles a space of distributions, so that one can define products, convolutions and Fourier transforms on the space (at least on dense subspaces). On the other hand side, it also has features related to algebraic geometry; there is e.g. a 'hard Lefschetz theorem' and 'Hodge-Riemann bilinear relations' for valuations. I will present a fairly general representation formula for translation invariant valuations, that gives an explanation of this (even though some proofs are still lacking).