The Transfer Principle of Integral Geometry and Isoperimetric Inequalities

经典积分几何中最显著的见解之一是霍华德的转移原理，它允许某些运动学公式在黎曼齐性空间之间逐字转移，只要空间具有相同的维数和同构的各向同性群。因此，例如，欧氏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。开创性的工作阿列斯克在理论的估值是起点的最新发展，积分几何。最近有人指出，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。在阿列斯克的理论框架内的价值观流形霍华德的转移原则变成猜想，在一定条件下代数不变的价值同构。Bernig、Fu和Solanes已经证实了这一点。该项目的一个重点是研究受限制的传输原理。特别是，它的有效性异常各向同性空间将被检查。经典积分几何与研究几何变分问题、内禀体积的等周不等式或拟质量积分有着密切的联系。这些不等式和基本的Aleksandrov-Fenchel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，这些不等式中的一些甚至直接来自运动学公式。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，为发现复向量空间中新的（并且可能非常有用的）不等式铺平了道路。以前发现的不平等现象似乎只是冰山一角。该项目的另一个目的是系统地研究和发现复向量空间中的新等周不等式。

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.