Classification problems for valuations on convex functions

Valuations on sets, that is, finitely additive functions, play a key part in many problems in geometry, and relations between various geometric quantitites can be encoded in transforms relating certain families of valuations. For example, the surface area of a convex body can be calculated by averaging the areas of its shadows in all directions in other words, the surface area is related to the areas of suitable lower dimensional projections. This result goes back to Cauchy in the three dimensional case, but higher dimensional versions of this formula exist for a variety of geometric quantities. A very simple explanation for the existence of such formulas was given by Hadwiger in his fundamental theorem on the classification of continuous and rigid motion-invariant valuations on convex bodies. He showed that any such valuation has to be a combination of the intrinsic volumes, which form a finite family of valuations in each dimension. In particular, many integral geometric formulas can naturally be interpreted as valuations of this type and therefore reduce to combinations of the intrinsic volumes. More recently, Semyon Alesker showed that for many families of symmetries the corresponding space of continuous invariant valuations on convex bodies is also generated by a finite set of valuations. In all of these cases, various integral geometrc formulas have been established, in particular on complex vector spaces. Approximately 15 years ago, the notion of valuation was extended from sets to functions by Monika Ludwig. As part of this generalisation, the intrinsic volumes were extended from convex bodies to convex functions, giving rise to the so-called functional intrinsic volumes, which share many of their properties, and for which a Hadwiger-type classification result exists. The project aims to obtain general characaterization results for valuations on convex functions that are invariant under other symmetry groups and to establish suitable representation formulas for these functionals. In particular, we propose to leverage these characterization results to obtain different representation formulas and to determine the transforms connecting these descriptions. We expect that this research will lead to new integral geometric formulas for valuations on convex functions as well as Monge-Ampère operators, which have been the main tool to construct invariant valuations. Our main focus will be on invariant valuations on complex vector spaces, which are well understood in the geometric setting, whereas there has only recently been a corresponding characterization result for valuations on convex functions.