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

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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 R²ⁿ must itself be centrally symmetric, with generalizations to some other types of symmetry.