  A closed contact manifold is called Besse when all its Reeb orbits are
closed, and Zoll when they have the same minimal period. In this paper, we
provide a characterization of Besse contact forms for convex contact spheres
and Riemannian unit tangent bundles in terms of $S^1$-equivariant spectral
invariants. Furthermore, for restricted contact type hypersurfaces of
symplectic Euclidean spaces, we give a sufficient condition for the Besse
property via the Ekeland-Hofer capacities.
