In a series of papers, the PI and her collaborators started the systematic study of valuations or finitely additive measures within affine geometric analysis. In Euclidean geometry, the concept of valuation has long been known to be of fundamental importance. A landmark result was Hadwiger's classification of rigid motion invariant, continuous valuations on convex bodies in 1950. However, classical affine surface area and centro-affine surface area are not continuous and therefore not characterized by Hadwiger's theorem. In joint papers, the PI and M. Reitzner obtained a classification of affine invariant, upper semicontinuous valuations and established the first characterization of affine surface area and centro-affine surface area. The main goal of the proposed research is to complete the classification of affine valuations and use the results and techniques developed for this classification to solve fundamental problems in affine geometric analysis.
The proposal is centered on fundamental open questions in affine geometric analysis. The basic objects are functions and operators in ordinary Euclidean space that are invariant or covariant under linear transformations. Within the last few years, a substantial amount of research was devoted to investigate in depth the relations between these functions and operators. There are numerous applications of these results and the proposed research to the asymptotic theory of Banach spaces, differential geometry, ordinary and partial differential equations, and even to seemingly unrelated fields like geometric tomography, image processing and information theory.
