  We provide exact expressions for the electrostatic energy of
uniformly-charged prolate and oblate spheroidal shells. We find that
uniformly-charged prolate spheroids of eccentricity greater than 0.9 have lower
Coulomb energy than a sphere of the same area. For the volume-constrained case,
we find that a sphere has the highest Coulomb energy among all spheroidal
shells. Further, we derive the change in the Coulomb energy of a
uniformly-charged shell due to small, area-conserving perturbations on the
spherical shape. Our perturbation calculations show that buckling-type
deformations on a sphere can lower the Coulomb energy. Finally, we consider the
possibility of counterion condensation on the spheroidal shell surface. We
employ a Manning-Oosawa two-state model approximation to evaluate the
renormalized charge and analyze the behavior of the equilibrium free energy as
a function of the shell's aspect ratio for both area-constrained and
volume-constrained cases. Counterion condensation is seen to favor the
formation of spheroidal structures over a sphere of equal area for high values
of shell volume fractions.
