  In the present paper we study the minimization of energy integrals on the
sphere with a focus on an interesting clustering phenomenon: for certain types
of potentials, optimal measures are discrete or are supported on small sets. In
particular, we prove that the support of any minimizer of the $p$-frame energy
has empty interior whenever $p$ is not an even integer. A similar effect is
also demonstrated for energies with analytic potentials which are not positive
definite. In addition, we establish the existence of discrete minimizers for a
large class of energies, which includes energies with polynomial potentials.
