Integral Geometry on Convex Functions
Quick Facts
Principal Investigator: Fabian Mussnig
Due to the PI's parental leave, the project lead was temporarily transferred to Monika Ludwig from 03/2023 to 09/2023.
Project Duration: 12/2022–11/2025
University: TU Wien
Abstract
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 Cauchy’s 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.
Researchers Supported by this Project
since 12/2022: Mohamed Abdeldjalil Mouamine
Publications Supported by this Project
Preprints
Additive kinematic formulas for convex functions | arXiv:2403.06697
jointly with Daniel Hug and Jacopo Ulivelli
Kubota-type formulas and supports of mixed measures | arXiv:2401.16371
jointly with Daniel Hug and Jacopo Ulivelli