  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.
