Isoperimetric type inequalities and Minkowski valuations in a complex vector space

自从Dehn解决了Hilbert的第三个问题，即关于多面体体积的初等定义的可能性以来，凸集上的赋值理论一直被积极地研究。近年来取得了很大进展，并取得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。该项目的第一部分致力于获得酉值的等周型不等式。在最近证明了一个Aleksandrov-Fichel型不等式之后，我们建议利用最优传输来研究涉及体积和酉值的等周型不等式。第二条主线是Minkowski和Blaschke估值的分类结果。最近，我们得到了一些Minkowski赋值的刻画结果，用不等式作为刻画性质，而不是一般线性群在凸体空间上的某个子群作用下的等价性。我们建议进一步探索作为刻画性质的不等式，借助于Petty投影和张不等式得到投影体算子的刻画结果。对于m维复向量空间中的Minkowski赋值，我们也期望得到U(M)-等变Minkowksi赋值的分类结果。对于复向量空间，我们进一步提出了对Blaschke赋值的研究。在这里，我们期望得到一个新的曲率图像的概念，这对于仿射几何领域的进一步发展将是重要的。

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.