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arXiv · 2603.27221

Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra

Annalisa Cesaroni·Matteo Novaga
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Abstract

We investigate the isoperimetric problem for the Voronoi cells of three-dimensional lattices. Using Selling parameters, we derive an explicit closed formula for the scale-invariant isoperimetric quotient $F$ in terms of six non-negative variables. We then analyse the local behaviour of $F$ at the most relevant lattice configurations: we prove that the body-centered cubic lattice (BCC) is a strict local minimiser of $F$ at fixed volume, whereas the face-centered cubic lattice (FCC) and the simple cubic lattice (SC) are not local minimisers. Then, we consider a family of lattices which interpolates between BCC and FCC, showing that BCC is the global minimiser of $F$ restricted to this family.

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Annalisa Cesaroni, Matteo Novaga. 2026-03-28. Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra. https://arxiv.org/abs/2603.27221

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