Valuations on Convex Functions

Valuations (or additive functions) are fundamental in geometry and have been studied there ever since Dehn`s solution to Hilbert`s Third Problem in 1901. Hadwiger`s fundamental theorem gives a complete classification of continuous and rigid motion-invariant valuations on compact, convex sets and thereby characterizes the intrinsic volumes. Valuations naturally arise in many problems. Applications in integral geometry and geometric probabilities are classic. Recently, valuations have also been used successfully in materials science, astronomy, and tomography. The notion of valuation was extended to function spaces about ten years ago (Ludwig: Advances in Mathematics 2011; American Journal of Mathematics 2012). Such spaces are of great importance in all parts of mathematical analysis. The aim of the project is a systematic investigation of valuations on convex functions. The very successful geometric valuation theory and the associated results in integral geometry are proposed to be transferred from compact, convex sets to convex functions. This new theory is closely linked to convex analysis and applications in this area are part of the project.

The description of important parameters of spaces of functions helps in the application of these spaces. In the project, this was achieved for spaces of convex functions, and initial steps were taken for discrete integer problems. Specifically, a Hadwiger theorem, i.e., a classification of all invariant and continuous, finite-additive measures, considering translations and rotations, was achieved. The corresponding functionals, the so-called functional intrinsic volumes, were discussed and represented in various ways. Furthermore, measure-valued functionals were classified.