arXiv subjects
Publications and source records attributed to 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. The final scalar inequality is verified by an exact computer-assisted certificate.
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.