AN EXTENSION OF SCHAFFER'S DUAL GIRTH CONJECTURE TO GRASSMANNIANS

作者：Faifman Dmitry^*

来源：Journal of Differential Geometry, 2012, 92(2): 201-220.

In this note we introduce a natural Finsler structure on convex surfaces, referred to as the quotient Finsler structure, which is dual in a sense to the inclusion of a convex surface in a normed space as a submanifold. It has an associated quotient girth, which is similar to the notion of girth defined by Schaffer. We prove the analogs of Schaffer's dual girth conjecture (proved by Alvarez-Paiva) and the Holmes-Thompson dual volumes theorem in the quotient setting. We then show that the quotient Finsler structure admits a natural extension to higher Grassmannians, and prove the corresponding theorems in the general case. We follow Alvarez-Paiva's approach to the problem, namely, we study the symplectic geometry of the associated co-ball bundles. For the higher Grassmannians, the theory of Hamiltonian actions is applied.

• 出版日期2012-10