Geometric minimization of highly symmetric potentials

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 un-derstanding of how they affect the vacuum expectation values. For illustration, we apply this method to the S4 and A4-symmetric three-Higgs-doublet models. We find that at least three recent phenomenological analyses of the A4-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.