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

TL;DR

The paper proves that the vertices of a regular cross-polytope uniquely solve optimal covering and polarization problems for antipodal point configurations on spheres in various dimensions.

Contribution

It establishes the uniqueness of the regular cross-polytope configuration as the optimal solution for covering and polarization problems in higher dimensions.

Findings

Regular cross-polytope vertices solve the covering problem for d ≥ 5.

They also uniquely solve the polarization problem for d ≥ 3.

This is the first such proof for the covering problem in dimensions d ≥ 5.

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 ($d\geq 3$).