Algebraic structures on translation invariant valuations

Abstract

The aim of this diploma thesis is to study several algebraic operations on the space of smooth translation invariant valuations. First we collect fundamental notions and results, mainly from convex geometry and the theory of valuations. These include the classical theorem of McMullen and Alesker's Irreducibility theorem. As a consequence of the latter, one obtains the often needed density of mixed volumes in the space of translation invariant valuations. Also, the representation of smooth valuations by differential forms and the representation of even and smooth valuations by Crofton measures is discussed. The representation by Crofton measures then allows us to define the Alesker-Fourier transform for even and smooth valuations. We continue by defining the product for mixed volumes and extending it to smooth valuations after proving continuity. Moreover, we study the Poincaré Duality which is induced by the product. Next, we define the convolution of even and smooth valuations and establish its relation to the product via the Alesker-Fourier transform. Afterwards, we use the representation by differential forms to extend the convolution to all smooth valuations. Finally, a proof of the fundamental theorem of algebraic integral geometry is presented, establishing a connection between kinematic formulas and the previously defined algebraic structures.