Valuations on convex bodies

The concept of valuation lies at the heart of geometry. A valuation is a function defined an convex sets that is additive with respect to unions and intersections. The volume is an example. Among the numerous further examples are the surface area and more generally the intrinsic volumes as well as the affine surface area, projection bodies, and intersection bodies. Valuations arise naturally in many problems. Applications in integral geometry and geometric probability are classical. More recently, the connection to problems of polytopal approximation has been established. For these applications, results an the classification of valuations are most useful. This is the central subject of this project. In recent years, there has been rapid progress within this area of research. So there are new methods and tools to address these classical problems. The results of this project will have applications in geometric tomography, in the Field of polytopal approximation and in Minkowski and Finsler geometry.