Hessian Inequalities and Extensions to Sobolev Spaces
Hessian Inequalities and Extensions to Sobolev Spaces
Principal Investigator: Fabian Mussnig
Project Duration: 10/2020–02/2023
Fellowship abroad at the University of Florence: 10/2020–09/2021
Return phase at TU Wien: 09/2022–02/2023
Foreign Host: Andrea Cianchi
Among all geometric figures in the plane, the circle has the property that it has the largest possible area for a given circumference. Similarly, in three-dimensional space, a sphere has the largest volume for a given surface area. In mathematics, this fact is generally expressed by the so-called isoperimetric inequality, which is a special case of even more general volume inequalities.
Recently, the concept of surface area and similar related characteristics has been extended from geometric figures or bodies to convex functions. This is known as Hessian valuations. The aim of the project is to generalize the volume inequalities which are known for geometric objects to the new Hessian valuations. Furthermore, both the Hessian valuations as well as the new inequalities are to be extended to an even larger class of functions. One anticipates that this will not only lead to new insights and applications for functions themselves, but also to new approaches to problems related to the classical volume inequalities.
Inequalities and counterexamples for functional intrinsic volumes and beyond | arXiv:2412.05001
jointly with Jacopo Ulivelli
Kubota-type formulas and supports of mixed measures | arXiv:2401.16371
jointly with Daniel Hug and Jacopo Ulivelli
Additive kinematic formulas for convex functions | arXiv:2403.06697
Canadian Journal of Mathematics, to appear.
jointly with Daniel Hug and Jacopo Ulivelli
American Journal of Mathematics 147 (2025), no. 4, 927–955.
jointly with Andrea Colesanti and Monika Ludwig
Advances in Mathematics 413 (2023), Art. 108832, 35 pp.
jointly with Andrea Colesanti and Monika Ludwig
The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures | arXiv:2111.05648
Calculus of Variations and Partial Differential Equations 61 (2022), no. 5, Art. 181, 37 pp.
jointly with Andrea Colesanti and Monika Ludwig
In: Convex Geometry: Cetraro, Italy 2021, Lecture Notes in Mathematics 2332, CIME Foundation subseries, pp. 19–78, Springer, Cham, 2023.
jointly with Monika Ludwig
Snapshots of modern mathematics from Oberwolfach 11/2022, 12 pp.
02/2023 1 Week Research Visit in Karlsruhe, Germany – Daniel Hug
Talk (60 min.): Integral geometry on convex functions
09/2022 Harmonic Analysis Methods in Geometric Tomography, ICERM, Providence, Rhode Island, USA
Invited Talk (45 min.): Functional Intrinsic Volumes | Video
09/2021 DMV-ÖMG Annual Conference 2021, Section Geometry and Topology, University of Passau, Germany (online)
Talk (25 min.): Intrinsic volumes on convex functions
09/2021 Convex, Integral and Stochastic Geometry - International Conference in Honour of Rolf Schneider and Wolfgang Weil, Bad Herrenalb, Germany
Talk (30 min.): Integral geometric formulas for functional intrinsic volumes
06/2021 CMS 75th+1 Anniversary Summer Meeting, Session on "New perspectives on the Brunn-Minkowski theory", Ottawa, Canada (online)
Talk (30 min.): Functional intrinsic volumes and Hadwiger's theorem for convex functions