Energy capacity inequalities via an action selector

An action selector associates, in a suitable way, to each compactly supported Hamiltonian on a symplectic manifold an action value of the Hamiltonian. Action selectors are known to exist for a broad class of symplectic manifolds. We show how the existence of an action selector leads to sharp energy capacity inequalities between the Gromov width, the Hofer–Zehnder capacity, and the displacement energy. We also obtain sharp lower bounds for the smallest action of a closed characteristic on contact type hypersurfaces.