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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.27221 |
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| _version_ | 1866915897084477440 |
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| author | Cesaroni, Annalisa Novaga, Matteo |
| author_facet | Cesaroni, Annalisa Novaga, Matteo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra Cesaroni, Annalisa Novaga, Matteo Metric Geometry Combinatorics 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. |
| title | Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2603.27221 |