Global truncated-octahedral isoperimetric conjecture

Determine whether the Archimedean truncated octahedron minimizes the isoperimetric quotient among all parallelohedra in three-dimensional Euclidean space.

Background

A parallelohedron is a convex polyhedron that tiles three-dimensional Euclidean space face-to-face by translations. The paper studies the isoperimetric quotient, defined as surface area divided by volume to the power 2/3, and proves that the Archimedean truncated octahedron is a strict local minimizer among parallelohedra, with a quadratic stability estimate modulo rotations and positive dilations.

The stated conjecture is global rather than local: it asks whether the Archimedean truncated octahedron minimizes the isoperimetric quotient over the entire class of three-dimensional parallelohedra. The paper’s main theorem establishes only local minimality near the truncated octahedron. The paper’s subsequent-developments note reports that the conjecture was later completely resolved by other work.