Isoperimetric type inequalities and Minkowski valuations in a complex vector space

点击翻译按钮获取中文摘要

The theory of valuations on convex sets has been actively studied since Dehn's solution of the Third Problem of Hilbert, on the possibility of an elementary definition for volume of polytopes. In recent years much progress has been made and new classification results and new structures on valuations have been obtained. In this project, we aim to contribute in these directions. The first part of the project is devoted to obtain isoperimetric type inequalities for unitary valuations. Having proved recently an Aleksandrov-Fenchel type inequality, we propose to study isoperimetric type inequalities involving the volume and a unitary valuation by using optimal transport.The second main line of the project concerns classification results for Minkowski and Blaschke valuations. Recently, we have obtained some characterization results for Minkowski valuations by using inequalities as a characterization property, instead of the usual equivariance under the action of some subgroup of the general linear group acting on the space of convex bodies. We propose to further explore inequalities as characterizing properties to obtain a characterization result for the projection body operator by means of the Petty projection and Zhang inequalities. Concerning Minkowski valuations in an m-dimensional complex vector space, we also expect to obtain classification results for U(m)-equivariant Minkowksi valuations. For complex vector spaces, we further propose a study of Blaschke valuations. There, we expect to obtain a new notion of curvature image, which would be important for further developments in the area of affine geometry.