Integral geometry on convex functions

A classical formula, which was first proved by Cauchy, states that the surface area of any convex body in three-dimensional space can be computed by averaging over the areas of its projections onto two- dimensional planes. Later, more general formulas in arbitrary dimensions were discovered which not only concern surface area but the so-called intrinsic volumes. These and other similar formulas are considered in the field of integral geometry. Among others, integral geometry was used to establish vector-valued analogs of the intrinsic volumes which admit interesting integral geometric formulas of their own. In addition, tensor-valued generalizations of the intrinsic volumes were found. Recently, intrinsic volumes were extended from convex bodies to convex functions. These new functional intrinsic volumes generalize their classical counterparts and share many of their properties. In particular, new functional versions of Cauchys surface area formula were found. The project aims to find further formulas of this type and to establish integral geometry on convex functions. Furthermore, we plan to extend this new theory to the vector- and tensor-valued case, where new meaningful operators need to be defined first. There, we also propose to characterize the newly found operators. One anticipates that this research will serve as a gateway to various additional results such as inequalities. Furthermore, since the classical operators have applications in fields like material science or medical imaging, also their potential functional versions are likely candidates for such applications.

Can the surface area of a geometric figure be determined from its shadows? An elegant answer to this question is provided by Cauchy's surface area formula, a classic result in integral geometry-a field in which, simply put, one averages geometric functionals. Cauchy's formula states that the surface area of a convex body can be calculated by averaging the areas of its shadows from all directions. One of the most important results of this project is a generalization of this formula that considers averages of shadows about a fixed axis. This generalization subsequently found numerous applications, including addressing questions about inequalities for functions, Minkowski problems, and extensions of vector-valued quantities (such as the center of mass) to functions. Vector-valued quantities, along with integral geometry, form the second main focus of this project. This involves vectors that are associated with geometric objects. One example is the center of mass, which stands out, among other things, by the fact that it rotates with the object when it is rotated and is also displaced when the object is displaced. Together with other quantities such as the Steiner point, the center of mass is essentially one of the only additive vector-valued quantities that move with the object. In this project, similar vectors have now been identified, though these are associated with convex functions. It turns out that there are also very natural vectors for functions that rotate with the function but remain unchanged when the function is displaced-a new phenomenon that does not occur with convex bodies. The vectors mentioned above were subsequently characterized based on their geometric properties. That is, they are the only geometric characteristics with these properties. Such results, in turn, find application in integral geometry, which highlights the close connection between the topics of this project.