The Transfer Principle of Integral Geometry and Isoperimetric Inequalities

经典积分几何最显著的见解之一是Howard的传递原理，该原理允许某些运动学公式在黎曼齐次空间之间逐字传递，只要这些空间具有相同的维度和同构的各向同性群。因此，例如，欧几里得空间的经典积分几何公式可以很容易地转移到球面和双曲空间。Alesker在赋值理论方面的开创性工作是积分几何最新发展的起点。最近有研究表明，积分几何公式中的常数只不过是赋值不变的代数的结构常数。在Alesker流形赋值理论的框架内，Howard传递原理变成了猜想，即在一定条件下，不变赋值的代数同构。Bernig、Fu和Solanes已经证实了复杂空间形式的这一点。本项目的一个重点是对猜想迁移原理的研究。特别是，它对于特殊的各向同性空间的有效性将被检验。经典积分几何与几何变分问题和关于内禀体积或拟积分的等周不等式的研究密切相关。这些不等式和基本的亚历克桑德罗夫-芬切尔不等式在数学的许多分支中都有应用和联系。在低维情况下，其中一些不等式甚至直接从运动学公式中推导出来。近年来积分几何的迅速发展，特别是复空间形式中运动学公式的完全确定，为在复向量空间中发现新的(可能非常有用的)不等式铺平了道路。之前发现的不平等似乎只是冰山一角。该项目的另一个目标是系统地研究和发现复向量空间中的新的等周不等式。

One of the most remarkable insights of classical integral geometry is the transfer principle of Howard, which allows certain kinematic formulas to be transferred verbatim between riemannian homogeneous spaces provided that the spaces have the same dimension and isomorphic isotropy groups. Thus, for example, the classical integral geometric formulas of euclidean space can be easily transferred to the sphere and hyperbolic space. The pioneering work of Alesker in the theory of valuations is the starting point for the latest developments in integral geometry. Very recently it was shown that the constants occurring in integral geometric formulas are nothing but structure constants of algebras of invariant valuations. Within the framework of Alesker's theory of valuations on manifolds Howard's transfer principle turns into the conjecture that under certain conditions the algebras of invariant valuations are isomorphic. Bernig, Fu and Solanes have confirmed this for complex space forms. One focus of this project is the investigation of the conjectured transfer principle. In particular, its validity for exceptional isotropic spaces will be examined. Classical integral geometry is closely linked to the study of geometric variational problems and isoperimetric inequalities for the intrinsic volumes or quermassintegrals. These inequalities and the fundamental Aleksandrov-Fenchel inequalities have applications in and links to numerous branches of mathematics. In low dimensions, some of these inequalities follow even directly from the kinematic formulas. The rapid development of integral geometry in recent years and, in particular, the complete determination of the kinematic formulas in complex space forms, have paved the way for the discovery of new (and potentially very useful) inequalities in complex vector spaces. The previously discovered inequalities seem to be only the tip of the iceberg. Another aim of the project is the systematic investigation and discovery of new isoperimetric inequalities in complex vector spaces.