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The Transfer Principle of Integral Geometry and Isoperimetric Inequalities

积分几何传递原理与等周不等式

基本信息

批准号：
289866435
负责人：
Professor Dr. Thomas Wannerer
金额：
--
依托单位：
Institut für Mathematik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/289866435?language=en
关键词：
Transfer Principle Integral Geometry Isoperimetric

项目摘要

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.

经典积分几何最引人注目的见解之一是霍华德的传递原理，它允许某些运动学公式在黎曼齐次空间之间逐字传递，前提是这些空间具有相同的维度和同构各向同性群。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。阿莱斯克在估价理论方面的开创性工作是积分几何最新发展的起点。最近，人们发现积分几何公式中出现的常数只不过是不变估值代数的结构常数。在阿莱斯克流形估值理论的框架内，霍华德的转移原理转化为在一定条件下不变估值的代数是同构的猜想。 Bernig、Fu 和 Solanes 已经针对复杂的空间形式证实了这一点。该项目的重点之一是对猜想转移原理的研究。特别是，将检查其对于特殊各向同性空间的有效性。经典积分几何与几何变分问题和内在体积或克马斯积分的等周不等式的研究密切相关。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并与之相关。在低维度中，其中一些不等式甚至可以直接从运动学公式得出。近年来积分几何的快速发展，特别是复杂空间形式中运动学公式的完整确定，为发现复杂向量空间中新的（并且可能非常有用的）不等式铺平了道路。先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杂向量空间中新的等周不等式。