MathDB

Truncated Octahedral Conjecture

Let P3\mathcal{P}_3 be the class of convex three-dimensional parallelohedra, namely convex polyhedra that tile R3\mathbb{R}^3 by translations. For every V>0V>0 and every P∈P3P\in\mathcal{P}_3 with Vol⁡(P)=V\operatorname{Vol}(P)=V, let TVT_V be the regular Archimedean truncated octahedron with volume VV. Then Area⁡(P)≥Area⁡(TV)\operatorname{Area}(P)\geq \operatorname{Area}(T_V), with equality if and only if PP is congruent to TVT_V.

References

Primary source

arXiv

Additional references

computer-assisted-proofconvex-geometrygeometryoptimization
Posted

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to prove the conjecture, but the proof has not yet been independently checked.

The conjecture says that the equal-volume truncated octahedron has the least surface area among space-filling convex polyhedra of the relevant translation-tiling class. It is attributed to Bezdek and concerns the restricted parallelohedron version of Kelvin’s broader foam problem.

Known results

  • In 2024, local minima had been determined for parallelepipeds and hexagonal prisms, while the remaining parallelohedron cases were under investigation.
  • Cesaroni and Novaga (March 2026) proved that the body-centered-cubic lattice, whose Voronoi cell is the truncated octahedron, is a strict local minimizer at fixed volume.
  • The same paper showed that the simple-cubic lattice is not stationary and the face-centered-cubic lattice is stationary but not locally minimizing.
  • They established global minimality only in a particular low-dimensional family interpolating between the body-centered-cubic and face-centered-cubic lattices.

September 2026 claimed proof

The preprint The Truncated Octahedral Conjecture claims a complete proof of Bezdek’s conjecture, with its final scalar inequality certified exactly by computer. This claim is unverified in the retrieved evidence.

Current status (as of September 2026): A preprint claims the full conjecture is proved, but independent confirmation is absent; earlier results establish only local or restricted-family minimality.

Sources

Solutions 0

No solutions have been posted yet.