Minimal volume in the parallelepiped range

Determine whether, for every dihedral angle \(\alpha\in[\arccos(1/3),\pi/2)\), the minimum volume among finite-volume convex equiangular hyperbolic polyhedra with common dihedral angle \(\alpha\) is attained by the equiangular parallelepiped \(C_\alpha\), equivalently the equiangular 4-prism.

Background

The paper proves that the regular hyperbolic tetrahedron uniquely minimizes volume among equiangular hyperbolic polyhedra when π/3≤α<arccos⁡(1/3)\pi/3\leq\alpha<\arccos(1/3), the range in which the regular tetrahedron exists. For larger angles, the regular tetrahedron degenerates and the equiangular parallelepiped CαC_\alpha, or equiangular 4-prism, has the smallest possible number of faces and is identified as the natural candidate for the volume minimizer.

The authors explain that their method does not settle this conjecture on [arccos⁡(1/3),π/2)[\arccos(1/3),\pi/2). They identify two unresolved technical requirements: obtaining sufficiently strong lower bounds for atoroidal components and extending the admissibility of the direct lateral edge surgeries in the graph-type case, followed by comparison of all ordinary nn-prisms with the 4-prism.

References

The preceding argument does not yet settle this conjecture beyond \alpha_0. Rather, the decomposition isolates two remaining tasks. For an atoroidal component one needs a lower bound strong enough to dominate \Vol(C_\alpha) throughout the new interval. In the graph-type case one must extend the admissibility of the direct lateral edge surgeries and then compare every ordinary n-prism with the 4-prism. Thus the present method gives a conditional reduction of the problem, but these two uniform comparisons have not been proved for all \alpha\in[\alpha_0,\pi/2). Is it true that, for all \arccos\frac13\le\alpha<\frac\pi2, the minimum volume among equiangular hyperbolic polyhedra with dihedral angle \alpha is attained by the equiangular parallelepiped C_\alpha?

— Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range  (2608.14223 - Egorov, 14 Aug 2026) in Section 7, “A remark on the parallelepiped range”