An extension of sch¨affer’s dual girth conjecture to grassmannians

Dmitry Faifman Tel-Aviv University

Differential Geometry mathscidoc:1609.10284

Journal of Differential Geometry, 92, (2), 201-220, 2012
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 Sch¨affer. We prove the analogs of Sch¨affer’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.
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@inproceedings{dmitry2012an,
  title={AN EXTENSION OF SCH¨AFFER’S DUAL GIRTH CONJECTURE TO GRASSMANNIANS},
  author={Dmitry Faifman},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914071900930821946},
  booktitle={Journal of Differential Geometry},
  volume={92},
  number={2},
  pages={201-220},
  year={2012},
}
Dmitry Faifman. AN EXTENSION OF SCH¨AFFER’S DUAL GIRTH CONJECTURE TO GRASSMANNIANS. 2012. Vol. 92. In Journal of Differential Geometry. pp.201-220. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914071900930821946.
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