2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70269An 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.Besides minor changes, an appendix describing the constructions of the known action selectors is added. 27 pages, 1 figure, latexDifferential GeometrySymplectic Geometrytext