Absolute minima of potentials of certain regular spherical configurations

We use methods of approximation theory to find the absolute minima on sphere potential spherical (2m−3)-designs with a non-trivial index 2m that are contained in union m parallel hyperplanes, m≥2, whose locations satisfy certain additional assumptions. The interaction between points is described by function dot product, which has positive derivatives orders 2m−2, 2m−1, and 2m. This includes case classical Coulomb, Riesz, logarithmic potentials as well completely monotone distance squared. illustrate this result showing minimum set vertices icosahedron unit S2 R3 attained at dual dodecahedron one for icosahedron. configuration 240 minimal vectors E8 root lattice normalized lie S7 R8 2160 we describe.