Lebesgue universal covering constant
Description of constant
$C_{13b} = a$ is the infimal area of a convex planar set $\Omega$ that can cover a congruent copy of every convex planar set of diameter $1$.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| \(\frac{\pi}{3}=1.0471975512\dots\) | Trivial | Follows from Jung’s theorem [Elek1994]. |
| \(\frac{\sqrt3}{2}=0.8660254038\dots\) | [Pal1920] | Regular hexagon circumscribing unit disk. |
| \(2-\frac{2}{\sqrt3}=0.8452994616\dots\) | [Pal1920] | Truncation of the hexagon via an inscribed regular dodecagon. |
| \(\le 0.844137708436\) | [Spr1936] | Removed a further tiny region |
| \(\le 0.844137708398\) | [Han1992] (corrected in [BBG2015]) | Removed two additional microscopic regions |
| \(\le 0.844115297128419059\dots\) | [BBG2015] | Computer-assisted geometry, with high-precision verification by Greg Egan. |
| \(\le 0.8440935944\) | [Gib2018] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $\frac{\pi}{4}=0.7853981634\dots$ | Trivial | Use unit disk |
| $0.8257$ | [Elek1994] | Use unit disk and equilateral triangle |
| $0.8271$ | [Elek1994] | Also use regular $3^j$-gons |
| $0.832$ | [BS2005] | Rigorous computer-aided search using a circle, equilateral triangle, and regular pentagon |
Additional comments and links
- The Blaschke selection theorem implies that a minimal convex cover exists. [Elek1994]
- It suffices to cover all constant-width \(1\) sets [Vre1981].
- Nonconvex variants were studied by Duff [Duf1980], and higher dimensional variants in [ABPR2025].
- Wikipedia entry for this problem.
- Quanta article, Nov 2018.
- See also: Moser’s worm problem.
References
- [ABPR2025] Arman, Andrii; Bondarenko, Andriy; Prymak, Andriy; Radchenko, Danylo. On asymptotic Lebesgue’s universal covering problem. arXiv:2512.04023 (2025). https://arxiv.org/abs/2512.04023
- [BBG2015] Baez, John C.; Bagdasaryan, Karine; Gibbs, Philip. The Lebesgue universal covering problem. Journal of Computational Geometry 6 (2015), no. 1, 288–299. Preprint: https://arxiv.org/abs/1502.01251 (Also available as a PDF from Baez’s webpage: https://math.ucr.edu/home/baez/covering.pdf)
- [BS2005] Brass, Peter; Sharifi, Mehrbod. A lower bound for Lebesgue’s universal cover problem. International Journal of Computational Geometry & Applications 15 (2005), 537–544. DOI: 10.1142/S0218195905001828.
- [Duf1980] Duff, G. F. D. A smaller universal cover for sets of unit diameter. C. R. Math. Rep. Acad. Sci. Canada 2 (1980), no. 1, 37–42. (PDF index page: https://mathreports.ca/volume-issue/vol-02-1980/vol-02-1-1980/)
- [Elek1994] Elekes, Gy. Generalized breadths, circular Cantor sets, and the least area UCC. Discrete & Computational Geometry 12 (1994), 439–449. DOI: 10.1007/BF02574391. (Open PDF: https://link.springer.com/content/pdf/10.1007/BF02574391.pdf)
- [Gib2018] Gibbs, Philip. An Upper Bound for Lebesgue’s Covering Problem. arXiv:1810.10089 (2018). https://arxiv.org/abs/1810.10089
- [Han1992] Hansen, H. C. Small universal covers for sets of unit diameter. Geometriae Dedicata 42 (1992), 205–213. DOI: 10.1007/BF00147549.
- [Pal1920] Pál, Gyula. Über ein elementares Variationsproblem. Danske Matematisk-Fysiske Meddelelser III, 2 (1920).
- [Spr1936] Sprague, Roland. Über ein elementares Variationsproblem. Matematiska Tidsskrift Ser. B (1936), 96–99.
- [Vre1981] Vrećica, S. A note on sets of constant width. Publications de L’Institut Mathématique 29 (1981), 289–291.
Contribution notes
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