We consider the Karamata (majorization) inequality and minimums of the majorization (M-sets) for f-energy potentials of m-point configurations with some constraints. In particular, we discuss for spheres the optimality of regular simplexes, describe M-sets with a small number of points, define spherical f-design and study their properties. Then we consider relations between the notions of f-designs and M-sets, $\tau$-designs, and two-distance sets.