A general version of Hadwiger's characterization theorem

Abstract

In this thesis we study valuations on the space K(V) of convex bodies in an n-dimensional Euclidean vector space V taking values in an Abelian semigroup A. In particular, we analyze the space of continuous, translation invariant, complex valuations. We will see that this space is a Banach space. There is a natural continuous action of the general linear group Gl(n) on the space of continuous, translation invariant, complex valuations and thus the subspace of i-homogeneous valuations becomes an SO(n) module. Hence, it is possible to decompose this space into a sum of irreducible representations of SO(n). This thesis contains the full proof of this statement and the proof of a reformulation which can be seen as a Hadwiger-type characterization of continuous, translation invariant and SO(n)-equivariant tensor valuations of degree i. The proof we present was established by S. Alesker, A. Bernig and F. Schuster.