Local stability for the truncated octahedral conjecture

Abstract

This note proves that the truncated octahedron of Archimedes is a strict local minimizer of the isoperimetric quotient among parallelohedra in three dimensions. This extends the corresponding result from the five-dimensional lattice-Voronoi subclass to the full ten-dimensional local parameter space. A six-zone parametrization and Schur's lemma reduce the Hessian to two positive definite $2\times 2$ matrices.

arXiv comment15 pages. This version clarifies coordinates and normalization, justifies regularity and Taylor expansion, and improves proof organization