赋范空间的商围长及Schäffer对偶围长猜想在Grassmannian空间的推广
The quotient girth of normed spaces, and an extension of Schäffer's dual girth conjecture to Grassmannians
摘要：
在本文中，我们引入了一个自然的Finsler结构的凸曲面，称为射影Finsler结构，这是对偶的意义上的凸曲面在赋范空间中的明显的包含。它有一个相关联的投射围长，这与Schäffer定义的围长概念相似。我们证明了Schäffer的对偶围长猜想（由Álvarez-Paiva证明）和Holmes-Thompson对偶体积定理在射影设置类似。证明了投射Finsler结构对高阶Grassmannian有自然扩张，并在一般情况下证明了相应的定理.我们遵循Álvarez-Paiva的方法来解决这个问题，即，我们研究了相关的co-ball丛的辛几何。对于高阶Grassmannian，应用Hamilton作用量理论。
英文摘要：
In this note we introduce a natural Finsler structure on convex surfaces, referred to as the projective Finsler structure, which is dual in a sense to the obvious inclusion of a convex surface in a normed space. It has an associated projective 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 projective setting. We then show that the projective 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.
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