Sharpness of the Cohn–Elkies bound for circle packings
Cohn and Elkies gave an upper bound for the density of sphere packings in $\mathbb{R}^d$ by means of an auxiliary function, and conjectured it to be sharp for $d = 1, 2, 8, 24$. They proved this for $d = 1$; Viazovska settled $d = 8$, and Cohn, Kumar, Miller, Radchenko and Viazovska $d = 24$. Such a function in the plane would have to vanish on the nonzero points of the hexagonal lattice, and its Fourier transform on those of the dual lattice. Here $\hat{f}(t) = \int f(x) e^{-2\pi i \langle x, t \rangle} \, dx$.
Problem. Is there a radial function $f : \mathbb{R}^2 \to \mathbb{R}$, admissible in the sense of Cohn and Elkies, with $f(x) \le 0$ for $|x| \ge 2$, $\hat{f}(t) \ge 0$ for all $t$, $\hat{f}(0) > 0$ and $f(0) / \hat{f}(0) = \sqrt{3} / 6$?
Source: H. Cohn and N. Elkies, New upper bounds on sphere packings I, Ann. of Math. 157 (2003). Listed as Problem 42 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CA
- Keywords
- circle packing, Fourier analysis, linear programming bounds, sphere packing
- Literature checked
- Nov. 3, 2026 by clemens-admin
References
@article{cohnelkies2003,
author = {Cohn, Henry and Elkies, Noam},
title = {New upper bounds on sphere packings {I}},
journal = {Annals of Mathematics (2)},
volume = {157},
number = {2},
pages = {689--714},
year = {2003},
doi = {10.4007/annals.2003.157.689}
}
@article{viazovska2017dim8,
author = {Viazovska, Maryna S.},
title = {The sphere packing problem in dimension 8},
journal = {Annals of Mathematics (2)},
volume = {185},
number = {3},
pages = {991--1015},
year = {2017},
doi = {10.4007/annals.2017.185.3.7}
}
@article{ckmrv2017dim24,
author = {Cohn, Henry and Kumar, Abhinav and Miller, Stephen D. and Radchenko, Danylo and Viazovska, Maryna},
title = {The sphere packing problem in dimension 24},
journal = {Annals of Mathematics (2)},
volume = {185},
number = {3},
pages = {1017--1033},
year = {2017},
doi = {10.4007/annals.2017.185.3.8}
}
@misc{sardari2021interpolation,
author = {Sardari, Naser Talebizadeh},
title = {Higher {F}ourier interpolation on the plane},
year = {2021},
eprint = {2102.08753},
archivePrefix = {arXiv}
}
@misc{openai2026circle,
author = {{OpenAI}},
title = {A sharp {F}ourier certificate for planar circle packing},
howpublished = {\url{https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf}},
note = {AI-generated, unrefereed; dated 23 September 2026},
year = {2026}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 42; version of January 2026, most recent update December 2025},
year = {2026}
}
Resolution claims
We resolve the planar Cohn--Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density $\pi/(2\sqrt3)$. We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-11-08 | Needs verification → Solution claimed | — | Recorded here: math @ 4eaee87, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01134, "Sharpness of the Cohn–Elkies bound for circle packings", clopen.solutions.