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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-polytopeinscribed in $S^{d-1}$ uniquely solves the best-covering problem (this is newfor $d\geq 5$) and the maximal polarization problem for potentials given by afunction of the distance squared with a positive and convex second derivative($d\geq 3$).

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