Isoperimetric type inequalities and Minkowski valuations in a complex vector space

自从Dehn解决了希尔伯特第三问题（关于多面体体积的初等定义的可能性）以来，凸集上的赋值理论得到了积极的研究。近年来，这一研究取得了很大进展，得到了新的分类结果和新的估值结构。在这个项目中，我们的目标是在这些方向上做出贡献。该项目的第一部分是致力于获得酉估值的等周型不等式。在最近证明了Aleksandrov-Fenchel型不等式之后，我们提出利用最优迁移研究包含体积和酉赋值的等周型不等式.第二条主线是关于Minkowski和Blaschke赋值的分类结果.最近，我们利用不等式作为Minkowski赋值的一个特征性质，而不是通常的在一般线性群的某个子群作用于凸体空间时的等方差，得到了Minkowski赋值的一些特征结果.我们建议进一步探讨不等式作为特征性质，通过Petty投影和张不等式得到投影体算子的一个特征结果。关于m维复向量空间中的Minkowski赋值，我们也期望得到U（m）-等变Minkowski赋值的分类结果.对于复向量空间，我们进一步研究了Blaschke赋值。在那里，我们期望得到一个新的概念的曲率图像，这将是重要的，在该地区的仿射几何的进一步发展。

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.