  These notes are devoted to various considerations on a family of sharp
interpolation inequalities on the sphere, which in dimension two and higher
interpolate between Poincar\'e, logarithmic Sobolev and critical Sobolev
(Onofri in dimension two) inequalities. We emphasize the connexion between
optimal constants and spectral properties of the Laplace-Beltrami operator on
the sphere. We shall address a series of related observations and give proofs
based on symmetrization and the ultraspherical setting.
