  In non-minimal Higgs mechanisms, one often needs to minimize highly symmetric
Higgs potentials. Here we propose a geometric way of doing it, which,
surprisingly, is often much more efficient than the usual method. By
construction, it gives the global minimum for any set of free parameters of the
potential, thus offering an intuitive understanding of how they affect the
vacuum expectation values. For illustration, we apply this method to the S_4
and A_4-symmetric three-Higgs-doublet models. We find that at least three
recent phenomenological analyses of the A_4-symmetric model used a local, not
the global minimum. We discuss coexistence of minima of different types, and
comment on the mathematical origin of geometrical CP-violation and on a new
symmetry linking different minima.
