Abstract
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 Schäffer. We prove the analogs of Schäffer’s dual girth conjecture (proved by Álvarez-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 Álvarez-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.
Citation
Dmitry Faifman. "An extension of Schäffer's dual girth conjecture to Grassmannians." J. Differential Geom. 92 (2) 201 - 220, October 2012. https://doi.org/10.4310/jdg/1352297806
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