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We use linear programming techniques to find points of absolute minimum overthe unit sphere $S^{d}$ in $\mathbb R^{d+1}$ of the total potential of a pointconfiguration $\omega_N\subset S^{d}$ which is a spherical $(2m-1)$-designcontained in the union of some $m$ parallel hyperplanes. The interactionbetween points is described by the kernel $K({\bf x},{\bf y})=f(\left|{\bfx}-{\bf y}\right|^2)$, where $\left|\ \!\cdot\ \!\right|$ is the Euclidean normin $\mathbb R^{d+1}$. The potential function $f$ is assumed to have a convexderivative $f^{(2m-2)}$. Points of minimum do not depend on $f$ and are thoseand 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 inJanuary 2022. Using this result, we find sets of universal minima of certainsix configurations on higher-dimensional spheres.

Absolute Minima of Potentials of a Certain Class of Spherical Designs