The Alesker-Poincaré duality

Abstract

The Alesker-Poincaré Duality is an embedding of the space of smooth translation invariant valuations on convex bodies into its dual space. It is required to formulate the Fundamental Theorem of Algebraic Integral Geometry, and is based on the Alesker product, an algebraic operation on the space of smooth translation invariant valuations. This thesis contains a collection of notions and proofs for this theory. Furthermore the convolution of smooth valuations allows for an analogue duality, which is treated in the thesis also. Finally there is an extension of the notion of valuations on convex bodies to valuations on manifolds, which allows for a similar treatment, resulting again in a duality map, a collection of this theory is the final part of this thesis.