Absolute Minima of Potentials of a Certain Class of Spherical Designs

We use linear programming techniques to find points of absolute minimum over the unit sphere $S^d$ in ${\mathbb{R}}^{d+1}$ of the total potential of a point configuration ${\omega }_N\subset S^d$ which is a spherical $(2m-1)$-design contained in the union of some $m$ parallel hyperplanes. The interaction between points is described by the kernel $K(\mathbf{x},\mathbf{y})=f({\mid \mathbf{x}-\mathbf{y}\mid }^2)$, where $\mid \text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}\mid$ is the Euclidean norm in ${\mathbb{R}}^{d+1}$. The potential function $f$ is assumed to have a convex derivative $f^{(2m-2)}$. Points of minimum do not depend on $f$ and are those and only those which form exactly $m$ distinct dot products with points of ${\omega }_N$. The proof of this theorem was presented at a workshop at ESI in January 2022. Using this result, we find sets of universal minima of certain six configurations on higher-dimensional spheres.