The Transfer Principle of Integral Geometry and Isoperimetric Inequalities

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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.