Lebesgue universal covering constant

Description of constant

$C_{13 b} = a$ is the infimal area of a convex planar set $Ω$ that can cover a congruent copy of every convex planar set of diameter $1$ .

Known upper bounds

Bound	Reference	Comments
$\frac{π}{3} = 1.0471975512 \dots$	Trivial	Follows from Jung’s theorem [Elek1994].
$\frac{3}{2} = 0.8660254038 \dots$	[Pal1920]	Regular hexagon circumscribing unit disk.
$2 - \frac{2}{3} = 0.8452994616 \dots$	[Pal1920]	Truncation of the hexagon via an inscribed regular dodecagon.
$\leq 0.844137708436$	[Spr1936]	Removed a further tiny region
$\leq 0.844137708398$	[Han1992] (corrected in [BBG2015])	Removed two additional microscopic regions
$\leq 0.844115297128419059 \dots$	[BBG2015]	Computer-assisted geometry, with high-precision verification by Greg Egan.
$\leq 0.8440935944$	[Gib2018]

Known lower bounds

Bound	Reference	Comments
$\frac{π}{4} = 0.7853981634 \dots$	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.

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.

Current state

The problem