We prove that for every subset of a tame symplectic manifold meeting a semi-positivity condition, the -sensitive Hofer--Zehnder capacity of is not greater than four times the stable displacement energy of ,
This estimate yields almost existence of periodic orbits near stably displaceable energy levels of time-independent Hamiltonian systems. Our main applications are: The Weinstein conjecture holds true for every stably displaceable hypersurface of contact type in . The flow describing the motion of a charge on a closed Riemannian manifold subject to a non-vanishing magnetic field and a conservative force field has contractible periodic orbits at almost all sufficiently small energies. The proof of the above energy-capacity inequality combines a curve shortening procedure in Hofer geometry with the following detection mechanism for periodic orbits: If the ray , , of Hamiltonian diffeomorphisms generated by a compactly supported time-independent Hamiltonian stops to be a minimal geodesic in its homotopy class, then a non-constant contractible periodic orbit must appear.