Max–min polarization optimality of the regular cross-polytope

Establish the max–min polarization optimality of the regular cross-polytope in fS^{n-1} for the relevant class of potentials, thereby resolving whether the cross-polytope maximizes the minimum value of the discrete potential among antipodal 2n-point spherical codes.

Background

The paper discusses max–min polarization, in which one seeks a configuration of N points on the sphere that maximizes the minimum value of a discrete potential over all points of the sphere. Regular polygons, simplices, and several other configurations are known to have this property for broad classes of potentials.

The regular cross-polytope is known to satisfy the max–min polarization property for centered codes, a class that includes all antipodal 2n-point codes. The unrestricted max–min polarization optimality of the cross-polytope itself is explicitly left unresolved.

References

While the max-min polarization property of the cross-polytope is still open, it was established by Boyvalenkov et. al. in Proposition 6.7 for all centered codes (i.e. codes for which there is a point on the sphere whose inner products to points of the cross-polytope are in the interval $[-1/\sqrt{n},1/\sqrt{n}]$; this class includes all antipodal $2n$-point codes).

— Universal polar dual pairs of spherical codes found in $E_8$ and $Λ_{24}$  (2512.25037 - Borodachov et al., 31 Dec 2025) in Section 1, Introduction