The three-dimensional lattice-periodic Kelvin problem
Let $L \subset \mathbb{R}^3$ range over full-rank lattices of covolume one, and let $E \subset \mathbb{R}^3$ range over bounded measurable sets of volume one and finite perimeter whose translates $\{E + \ell : \ell \in L\}$ partition $\mathbb{R}^3$ up to sets of Lebesgue measure zero. Here $\operatorname{Per}(E)$ is the total variation of the distributional gradient of $1_E$, and $$\mathcal{K}_{\mathrm{lat},3} = \inf_{(L,E)} \frac{1}{2} \operatorname{Per}(E),$$ the factor $1/2$ counting each interface between cells once. Curved interfaces and nonconvex cells are allowed. Kelvin proposed a slightly modified truncated octahedron, which tiles space by lattice translates, as the cell of least area in a partition of space into cells of unit volume (1887). Weaire and Phelan found a partition into cells of equal volume, of two different shapes, with less area than Kelvin's (1994); it is not a lattice tiling. Cesaroni and Novaga proved, in every dimension, that the corresponding infimum over lattice tilings is attained and that minimal cells are bounded (2024). In the plane the minimal cell is a regular hexagon, by Hales' honeycomb theorem (2001). Among convex cells, which tile space by lattice translates only if they are parallelohedra, the regular truncated octahedron uniquely minimizes surface area at fixed volume, as Hales and Song, and independently Cesaroni and Novaga, proved (2026).
Problem. Determine the exact value of $\mathcal{K}_{\mathrm{lat},3}$ and characterize the minimizing pairs $(L, E)$, if the infimum is attained.
Source: Thomson (Lord Kelvin) (1887), for partitions of space into cells of equal volume; in this lattice-periodic form, see Cesaroni and Novaga (2024, 2025). Stated by AIM — Open Applied Problems as the determination of the minimal value and the minimizers.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG, MG
- Keywords
- AIM Open Applied Problems, Kelvin problem, lattice tilings, minimal partitions
References
@article{thomson1887,
author = {Thomson, William},
title = {On the division of space with minimum partitional area},
journal = {Philos. Mag. (5)},
volume = {24},
number = {151},
year = {1887},
pages = {503--514},
doi = {10.1080/14786448708628135},
}
@incollection{cesaroni2024,
author = {Cesaroni, Annalisa and Novaga, Matteo},
title = {Minimal periodic foams with equal cells},
booktitle = {Anisotropic Isoperimetric Problems and Related Topics},
editor = {Franceschi, Valentina and Pluda, Alessandra and Saracco, Giorgio},
series = {Springer INdAM Series},
publisher = {Springer},
address = {Singapore},
year = {2024},
pages = {15--24},
doi = {10.1007/978-981-97-6984-1_2},
eprint = {2302.07112},
archivePrefix = {arXiv},
}
@article{cesaroni2025,
author = {Cesaroni, Annalisa and Novaga, Matteo},
title = {Minimal periodic foams with fixed inradius},
journal = {Mathematika},
volume = {71},
number = {2},
year = {2025},
pages = {e70020},
doi = {10.1112/mtk.70020},
eprint = {2407.07534},
archivePrefix = {arXiv},
}
@article{weaire1994,
author = {Weaire, D. and Phelan, R.},
title = {A counter-example to {K}elvin's conjecture on minimal surfaces},
journal = {Philos. Mag. Lett.},
volume = {69},
number = {2},
year = {1994},
pages = {107--110},
doi = {10.1080/09500839408241577},
}
@article{hales2001,
author = {Hales, Thomas C.},
title = {The honeycomb conjecture},
journal = {Discrete Comput. Geom.},
volume = {25},
number = {1},
year = {2001},
pages = {1--22},
doi = {10.1007/s004540010071},
}
@misc{hales2026,
author = {Hales, Thomas and Song, Lark},
title = {The truncated octahedral conjecture},
year = {2026},
eprint = {2609.00997},
archivePrefix = {arXiv},
}
@misc{cesaroni2026,
author = {Cesaroni, Annalisa and Novaga, Matteo},
title = {The truncated octahedron minimizes surface area among parallelohedra of equal volume},
year = {2026},
eprint = {2609.02384},
archivePrefix = {arXiv},
}
@misc{aim-kelvin-lattice-foam,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The three-dimensional lattice-periodic Kelvin problem}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/084-kelvin-lattice-foam.md},
note = {Entry 084 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00605, "The three-dimensional lattice-periodic Kelvin problem", clopen.solutions.