Optimal Antipodal Configuration of $2d$ Points on a Sphere in $\mathbb R^d$ for Covering

Abstract

We show that among antipodal $2d$-point configurations on the sphere $S^{d-1}$ in $\mathbb R^d$, the set of vertices of a regular cross-polytope inscribed in $S^{d-1}$ uniquely solves the best-covering problem (this is new for $d\geq 5$) and the maximal polarization problem for potentials given by a function of the distance squared with a positive and convex second derivative ($d\geq 3$).