Gaussian-energy optimality of the triangular lattice among periodic configurations
Let $\Lambda_\triangle \subset \mathbb{R}^2$ be the triangular lattice scaled to covolume one. For an integer $N \ge 1$, a lattice $L \subset \mathbb{R}^2$ of covolume $N$ and points $a_1, \dots, a_N$ distinct modulo $L$, the periodic configuration $P = \bigcup_i (a_i + L)$ has point density one. Montgomery proved that the triangular lattice minimizes Gaussian energy among planar lattices of the same density, which is the case $N = 1$ below. Cohn and Kumar conjectured that in dimension two the triangular lattice, and in dimensions $8$ and $24$ the $E_8$ and Leech lattices, have the least energy among periodic configurations of their density for every completely monotone potential of squared distance with suitable decay, certified by a sharp Fourier auxiliary function. Cohn, Kumar, Miller, Radchenko and Viazovska proved the universal optimality of $E_8$ and the Leech lattice.
Problem. Is $$\frac{1}{N} \sum_{i,j=1}^{N} \ \sum_{\substack{v \in L \\ (i,j,v) \ne (i,i,0)}} e^{-\pi\alpha |a_i - a_j - v|^2} \ \ge \sum_{v \in \Lambda_\triangle \setminus \{0\}} e^{-\pi\alpha |v|^2}$$ for every $\alpha > 0$ and every such $N$, $L$ and $a_1, \dots, a_N$? In the left sum only the self-interactions $j = i$, $v = 0$ are omitted.
Source: Cohn and Kumar (2007), Conjecture 9.4, in dimension two; stated for Gaussian potentials as an energy inequality by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Gaussian energy, triangular lattice, universal optimality
References
@article{cohn2007,
author = {Cohn, Henry and Kumar, Abhinav},
title = {Universally optimal distribution of points on spheres},
journal = {J. Amer. Math. Soc.},
volume = {20},
number = {1},
year = {2007},
pages = {99--148},
doi = {10.1090/S0894-0347-06-00546-7},
eprint = {math/0607446},
archivePrefix = {arXiv},
}
@article{montgomery1988,
author = {Montgomery, Hugh L.},
title = {Minimal theta functions},
journal = {Glasgow Math. J.},
volume = {30},
number = {1},
year = {1988},
pages = {75--85},
doi = {10.1017/S0017089500007047},
}
@article{cohn2022,
author = {Cohn, Henry and Kumar, Abhinav and Miller, Stephen D. and Radchenko, Danylo and Viazovska, Maryna},
title = {Universal optimality of the {$E_8$} and {L}eech lattices and interpolation formulas},
journal = {Ann. of Math. (2)},
volume = {196},
number = {3},
year = {2022},
pages = {983--1082},
doi = {10.4007/annals.2022.196.3.3},
eprint = {1902.05438},
archivePrefix = {arXiv},
}
@misc{aim-triangular-universal-optimality,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Gaussian-energy optimality of the triangular lattice among periodic configurations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/087-triangular-universal-optimality.md},
note = {Entry 087 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00730, "Gaussian-energy optimality of the triangular lattice among periodic configurations", clopen.solutions.