Integral Geometry on Convex Functions
Integral Geometry on Convex Functions
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
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.
since 12/2022: Mohamed Abdeldjalil Mouamine
Mohamed Abdeldjalil Mouamine
Additive kinematic formulas for functional Minkowski vectors
arXiv:2508.07984
Mohamed Abdeldjalil Mouamine, Fabian Mussnig
The vectorial Hadwiger theorem on convex functions
arXiv:2504.04952
Mohamed Abdeldjalil Mouamine, Fabian Mussnig
A Klain-Schneider theorem for vector-valued valuations on convex functions
arXiv:2503.07287
Fabian Mussnig, Jacopo Ulivelli
Inequalities and counterexamples for functional intrinsic volumes and beyond
arXiv:2412.05001
Daniel Hug, Fabian Mussnig, and Jacopo Ulivelli
Kubota-type formulas and supports of mixed measures
arXiv:2401.16371
Daniel Hug, Fabian Mussnig, and Jacopo Ulivelli
Additive kinematic formulas for convex functions
Canadian Journal of Mathematics, to appear.
arXiv:2403.06697
Andrea Colesanti, Monika Ludwig, and Fabian Mussnig
The Hadwiger theorem on convex functions, I
Geometric and Functional Analysis 34 (2024), no. 6, 1839–1898.
arXiv:2009.03702
05/2024 Conference on Convex Geometry and Related PDEs, Changsha, China
Invited Talk (45 min.): Integral geometry and supports of mixed measures