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

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.