The quotient girth of normed spaces, and an extension of Sch\"affer's dual girth conjecture to Grassmannians

TL;DR

This paper introduces a new Finsler structure on convex surfaces, extends Sch"affer's dual girth conjecture to Grassmannians, and proves related theorems using symplectic geometry and Hamiltonian actions.

Contribution

It extends the dual girth conjecture and Holmes-Thompson theorem to a projective Finsler structure on convex surfaces and higher Grassmannians.

Findings

Proved analogs of Sch"affer's dual girth conjecture in the projective setting.

Extended the theory to higher Grassmannians with corresponding theorems.

Applied symplectic geometry and Hamiltonian actions to study the structure.

Abstract

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 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 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 \'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.