Polarization Problem on a Higher-Dimensional Sphere for a Simplex

We study the problem of maximizing minimal value over sphere $$S^{d-1}\subset {\mathbb {R}}^d$$ potential generated by a configuration $$d+1$$ points on $$S^{d-1}$$ (the maximal discrete polarization problem). The interact via given function f Euclidean distance squared, where $$f:[0,4]\rightarrow (-\infty ,\infty ]$$ is continuous (in extended sense), decreasing [0, 4], and finite convex (0, with concave or derivative $$f'$$ . prove that vertices regular d-simplex inscribed in optimal. This result new for $$d>3$$ (certain special cases $$d=2$$ $$d=3$$ are also new). As byproduct, we find simpler proof known optimal covering property