Algebraic integral geometry for convex functions

Valuations are a classical instrument which allows the study of the geometric properties of an object. Suppose the object in question is convex. That is, it contains every segment joining every couple of points in the object. In this case, such geometric descriptors can be classified under suitable topological and geometric assumptions. A notable example is Hadwiger`s theorem, which characterizes all continuous valuations on convex bodies invariant under translations and rotations. The theorem states that intrinsic volumes span this space of valuations. These are functionals which are the staples of the so- called Brunn-Minkowski theory. Hadwiger`s theorem has applications, for example, in integral geometry, leading to kinematic and Kubota-type formulas. More recently, Alesker, Bernig, and Fu have initiated the study of algebraic structures on spaces of valuations, which has led to substantial advancements and new interpretations of the aforementioned integral-geometric formulas. Later, Colesanti, Ludwig, and Mussnig introduced the theory of valuations on convex functions, which extended the theory of valuations on convex bodies. In this framework, the study of algebraic structures on spaces of valuations on convex functions is an untapped line of research that this project aims to pursue. We plan to introduce algebraic operations on these functional spaces and study the structure that arises from them. The final aim is to connect these structures with integral geometric formulas, as it happens in the geometric case. This project interacts with many different fields of mathematics, such as geometric measure theory, algebraic geometry, and calculus of variations, and we expect them to have substantial interplay with our results.