Our main interest will be studying and participating on the development of modern aspects of the so called Brunn-Minkowski theory [Sh]. This branch of mathematical analysis and integral geometry is mainly focused on the concept of convex bodies and their properties, with results that touch both applications and theory.
One of the most recent developments is the concept of valuations of functions. This is a generalization of a celebrated tool of the classical Brunn-Minkowski theory, namely valuations on the lattices of convex sets, which through the Hadwiger's characterization Theorem [Ha] allows many useful integral representation for Quermassintegrals and Intrinsic volumes. In the new context of convex functions, valuations have been studied in particular by Monika Ludwig [L1],[CLM1-3], with whom we will work during a planned stay in Vienna. These new valuations are deeply connected with Monge-Ampere equations through Hessian measures, a concept introduced by Trudinger and Wang [TW] in the study of Monge-Ampere equations. This connection suggests the underlying role of Optimal transportation, which will be one of the aspect we will focus on.
Parallely we will study the role of symmetrizations in the context of valuations. Symmetrizations are maps on families of sets which transform a given object in a more symmetrical element of the same family preserving certain properties of the given set. In general the preserved quantities for convex sets are intrinsic volumes. Symmetrizations have been widely used, especially in the field of PDEs, to obtain estimates and inequalities, as the celebrated Faber-Krahn inequality [He], which is obtained using the Schwarz symmetrization of the epigraph of a function. Nowdays this topic has been deeply studied by Bianchi, Gardner and Gronchi [BGG1],[BGG2], who obtained many characterization and classification results on symmetrizations, inspiring our recent works [Ul].
The modern theory of valuations is a quite recent field, which has been developed in the last twenty years and started with the works of Alesker. Though very important results have been obtained following the methods of classical convex geometry, we hope that the use of Optimal Transport and symmetrization techniques could lead to new sharp results and understanding of these valuations. Again, it will be crucial to spot suitable valuations compatible with this instruments. The striking links that both optimal transport and Hessian valuations have with Monge-Ampere operators suggest that this approach could strengthen the applications to convexity of transport techniques.