AI 中文总结
该研究证明三维欧氏空间中阿基米德截角正八面体是平行多面体等周商的严格局部极小值，通过相关方法扩展了对应结果的适用空间维度，并将海森矩阵简化为两个正定2×2矩阵。
AI 中文摘要
我们证明了阿基米德截角正八面体是三维欧氏空间中所有平行多面体的等周商严格局部极小值。经过体积归一化和旋转极小化后，亏缺量控制了该阿基米德胞的平方豪斯多夫距离。这将相应结果从五维晶格-沃罗诺伊子类扩展到完整的十维局部参数空间。六区域参数化与舒尔引理将海森矩阵简化为两个正定2×2矩阵。
英文摘要
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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