The minimum-area universal cover of planar diameter-one sets
Let $\mathcal{U}$ be the class of compact convex sets $K \subset \mathbb{R}^2$ such that every nonempty compact set $E \subset \mathbb{R}^2$ with $\operatorname{diam} E \le 1$ satisfies $g(E) \subseteq K$ for some Euclidean isometry $g$; reflections are allowed as well as rotations and translations. Let $$a_{\mathrm{Leb}} = \inf_{K \in \mathcal{U}} \operatorname{Area}(K).$$ Pál proved that the regular hexagon circumscribed about a disc of diameter $1$ belongs to $\mathcal{U}$, so $a_{\mathrm{Leb}} \le \sqrt{3}/2$, and cut two corners off it to reach $2 - 2/\sqrt{3} = 0.8452\ldots$. After further cuts by Sprague and Hansen, Baez, Bagdasaryan and Gibbs constructed a universal cover of area less than $0.8441153$, and Gibbs one of area $0.8440935944$. Brass and Sharifi proved $a_{\mathrm{Leb}} \ge 0.832$; 2026 preprints of Zeng and of Mishra raise the lower bound to $0.834$ and $0.8344$.
Problem. Determine $a_{\mathrm{Leb}}$ exactly, and characterize the minimizing covers. The sets to be covered are arbitrary sets of diameter at most $1$, not only curves of prescribed length.
Source: Lebesgue (1914); see Pál (1920) and Baez, Bagdasaryan and Gibbs (2015), §1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, sets of constant width, universal covers
References
@article{pal1920,
author = {P{\'a}l, Julius},
title = {{\"U}ber ein elementares {V}ariationsproblem},
journal = {Danske Vid. Selsk. Math.-Fys. Medd.},
volume = {3},
number = {2},
year = {1920},
pages = {1--35},
}
@article{baez2015,
author = {Baez, John C. and Bagdasaryan, Karine and Gibbs, Philip},
title = {The {L}ebesgue universal covering problem},
journal = {J. Comput. Geom.},
volume = {6},
number = {1},
year = {2015},
pages = {288--299},
doi = {10.20382/jocg.v6i1a12},
eprint = {1502.01251},
archivePrefix = {arXiv},
}
@misc{gibbs2018,
author = {Gibbs, Philip},
title = {An upper bound for {L}ebesgue's covering problem},
year = {2018},
eprint = {1810.10089},
archivePrefix = {arXiv},
}
@article{brass2005,
author = {Brass, Peter and Sharifi, Mehrbod},
title = {A lower bound for {L}ebesgue's universal cover problem},
journal = {Internat. J. Comput. Geom. Appl.},
volume = {15},
number = {5},
year = {2005},
pages = {537--544},
doi = {10.1142/S0218195905001828},
}
@misc{zeng2026,
author = {Zeng, Shuai},
title = {An exact hierarchy for {L}ebesgue's universal covering constant and a certified 0.834 lower bound},
year = {2026},
eprint = {2609.01284},
archivePrefix = {arXiv},
}
@misc{mishra2026,
author = {Mishra, Ujjwal},
title = {Curves of constant width and {L}ebesgue's covering problem},
year = {2026},
eprint = {2608.30538},
archivePrefix = {arXiv},
}
@misc{aim-lebesgue-universal-cover,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The minimum-area universal cover of planar diameter-one sets}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/262-lebesgue-universal-cover.md},
note = {Entry 262 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00532, "The minimum-area universal cover of planar diameter-one sets", clopen.solutions.