Wider research context/theoretical framework
Among the central results of the Brunn-Minkowski theory are the isoperimetric and Brunn-Minkowski inequalities for intrinsic volumes as well as the Aleksandrov-Fenchel inequality for mixed volumes. On the other hand, Hessian measures play a crucial role in the study of partial differential equations and Sobolev spaces. Recently, Hessian measures were used to establish new important functional analogs of intrinsic and mixed volumes for convex functions.
Hypotheses/research questions/objectives
The proposed research aims to establish new functional counterparts of isoperimetric, Brunn-Minkowski and Aleksandrov-Fenchel inequalities for convex functions. Furthermore, we plan to extend this theory to Sobolev spaces. We intend to investigate the equality cases of the new inequalities. In particular, we propose the use of analytic tools to attack the still open equality cases of the classical inequalities.
Approach/methods
Based on the already obtained results for the new functional versions of the intrinsic and mixed volumes, we plan to use several advanced techniques in order to obtain inequalities. Moreover, we want to use different extensions of Hessian measures to Sobolev spaces such as the recently introduced distributional k-Hessians for fractional Sobolev spaces.
Level of originality/innovation
The basis of the functional setting for this project is very new and was only recently introduced by the applicant together with Andrea Colesanti and Monika Ludwig. In particular, no inequalities were previously established and there are no known extensions of the new functionals to Sobolev spaces.
Primary researchers involved
The applicant plans to use the Erwin Schrödinger fellowship to work at the University of Florence for one year, followed by a 6 month return phase in Austria. In Florence, he will work with Prof. Andrea Cianchi who is a distinguished expert on Sobolev spaces and geometric aspects of partial differential equations. Furthermore, the applicant will collaborate with local professors Andrea Colesanti and Paolo Salani, who both made significant contributions to Brunn-Minkowski inequalities and Hessian capacities.
