In this thesis we investigate Blaschke Minkowski homomorphisms, i.e. continuous, rotation intertwining and Blaschke Minkowski mixed additive maps. One of the main results shows that these operators admit a representation via a spherical convolution operator. Moreover we give a complete classification of all even Blaschke Minkowski homomorphisms. The most widely known example of these maps is the projection body operator. The established results show that general Blaschke Minkowski homomorphisms behave in many respects similar to this prototype.
Motivated by important volume inequalities for projection bodies we study the behavior of the volume (and more general quermassintegrals) of the images of Blaschke Minkowski homomorphisms. We show that these operators behave similar to the volume functional with respect to Minkowski linear combinations and that for fundamental inequalities of the Brunn Minkowski theory there are analogous inequalities satisfied by the volume of the images of Blaschke Minkowski homomorphisms.
In recent years a theory for star bodies dual to the Brunn Minkowski Theory of convex bodies was developed. For inequalities of the classical theory of convex bodies there are analogous relations (often easier to prove) satisfied by star bodies. For many of our results there are corresponding dual counterparts.
Motivated by properties of the well known intersection body operator, the dual to the projection body operator, we define radial Blaschke Minkowski homomorphisms. We give a complete classification of these operators and show that they satisfy volume inequalities analogous to the inequalities we proved for Blaschke Minkowski homomorphisms.