Fejes–Tóth maximal-volume inscribed polyhedra

Determine the exact maximal volume C_toth(n) and characterize the extremal polyhedra with n vertices inscribed on the unit sphere S^2 for n > 8.

Background

The problem seeks the maximum volume of polyhedra with prescribed numbers of vertices on S2. Cases up to n = 8 have been resolved by geometric and computational methods; larger n exhibit complex combinatorics.

A full solution would unify geometric extremal principles and computational evidence for high-vertex configurations.

References

The case $n > 8$ remains open and the most up to date database of current optimal polytopes is maintained by Sloane .

— Mathematical exploration and discovery at scale  (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Points on sphere maximizing the volume” (Section 4.20)