  We consider convex contact spheres $Y$ all of whose Reeb orbits are closed.
Any such $Y$ admits a stratification by the periods of closed Reeb orbits. We
show that $Y$ "resembles" a contact ellipsoid: any stratum of $Y$ is an
integral homology sphere, and the sequence of Ekeland-Hofer spectral invariants
of $Y$ coincides with the full sequence of action values, each one repeated
according to its multiplicity.
