Valuations on Function Spaces

The concept of valuation lies at the heart of geometry. A valuation is a function defined on sets that is additive with respect to unions and intersections. The volume is an example. Among the numerous further examples are surface area and more generally the intrinsic volumes as well as 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, valuations have found important applications within Material Sciences, Astronomy and Tomography. The notion of valuation was recently extended to function spaces (Ludwig: Advances in Mathematics 2011; American Journal of Mathematics 2012). Such spaces are important in all parts of analysis and are the basic tool for many applied problems. Already now it is clear that some fundamental operators are characterized by the valuation property and basic invariance and continuity properties. The aim of the proposed research is to systematically study valuations on function spaces and to obtain basic classification theorems for such valuations. Such results will find applications in Geometric Analysis and more applied fields.

The concept of valuation lies at the heart of geometry. A valuation is a function defined on sets that is additive with respect to unions and intersections. The volume is an example. Among the numerous further examples are surface area and more generally the intrinsic volumes as well as 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, valuations have found important applications within Materials Science, Astronomy and Tomography. The notion of valuation was recently extended to function spaces (Ludwig: Advances in Mathematics 2011; American Journal of Mathematics 2012). Such spaces are important in all parts of analysis and are the basic tool for many applied problems. The aim of the project is to systematically study valuations on function spaces and to obtain basic classification theorems for such valuations. Such results were established for functions of bounded variation (Tuo Wang: Indiana University Mathematics Journal 2014) and for convex functions (Colesanti, Ludwig, and Mussnig: Calculus of Variations and PDE 2017, International Mathematics Research Notices, in press). The new operators on these spaces are analogues of the classical notion of volume and of the projection body. In addition, a complete classification of the important Laplace transform was established (Jin Li und Dan Ma: Journal of Functional Analysis 2017) and thus the first steps were made towards a theory of function-valued valuations. Also discrete valuations, in particular, valuations on lattice polytopes were classified (Ludwig und Silverstein: Advances in Mathematics 2017).