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math.MG — Metric Geometry

The truncated octahedral conjecture

Contributed by Thomas Hales, Lark Song

Among three-dimensional parallelohedra of fixed volume, the Archimedean truncated octahedron uniquely minimizes surface area, affirming a conjecture of K. Bezdek from 2006. One inequality in the proof is verified by an exact computer-assisted certificate. The simplex, frame, and graphical-zonotope inequalities used in the proof hold in arbitrary dimension.

A mix of human-written and AI-generated textHuman understanding: all partscomputer-assisted proofisoperimetric inequalitytranslational tiling

Oct 12, 2026PDF ↗View work →

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The truncated octahedral conjecture

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