Local quadratic isoperimetric stability of the truncated octahedron among parallelohedra
We prove that the Archimedean truncated octahedron is a strict local minimizer of the isoperimetric quotient among all parallelohedra in $\mathbb{R}^3$. After volume normalization and minimization over rotations, the deficit controls the squared Hausdorff distance from the Archimedean cell. 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.
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