Estimating symplectic capacities from lengths of closed curves on the unit spheres

TL;DR

This paper enhances estimates for symplectic capacities of convex bodies by demonstrating that minimal action closed characteristics are centrally symmetric, with implications for understanding symplectic invariants and geometric symmetries.

Contribution

It introduces new symmetry properties of minimal action closed characteristics on convex bodies, improving capacity estimates and extending symmetry results.

Findings

Minimal action closed characteristics are centrally symmetric.

Improved estimates for Ekeland--Hofer--Zehnder capacity.

Generalizations to other symmetry types.

Abstract

We improve the estimates for the Ekeland--Hofer--Zehnder capacity of convex bodies by Gluskin and Ostrover. In the course of our argument we show that a closed characteristic of minimal action on the boundary of a centrally symmetric convex body in $\mathbb R^{2n}$ must itself be centrally symmetric, with generalizations to some other types of symmetry.