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OP-12015OpenGeometry and topology

Lebesgue's convex universal cover constant (C13bC_{13b})

Lebesgue's convex universal cover constant: determine C13bC_{13b} or improve its bounds; the repository records 0.832 (lower) and 0.8440935944 (upper).

MSC 52A40 · Inequalities and extremum problems involving convexity in convex geometry

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

C13b=aC_{13b} = a is the infimal area of a convex planar set Ω\Omega that can cover a congruent copy of every convex planar set of diameter 11.

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Background

References (from the repository page)

  • [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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.832; best known upper bound 0.8440935944. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
π3=1.0471975512…\frac{\pi}{3}=1.0471975512\dots Trivial Follows from Jung’s theorem [Elek1994].
32=0.8660254038…\frac{\sqrt3}{2}=0.8660254038\dots [Pal1920] Regular hexagon circumscribing unit disk.
2−23=0.8452994616…2-\frac{2}{\sqrt3}=0.8452994616\dots [Pal1920] Truncation of the hexagon via an inscribed regular dodecagon.
≤0.844137708436\le 0.844137708436 [Spr1936] Removed a further tiny region
≤0.844137708398\le 0.844137708398 [Han1992] (corrected in [BBG2015]) Removed two additional microscopic regions
≤0.844115297128419059…\le 0.844115297128419059\dots [BBG2015] Computer-assisted geometry, with high-precision verification by Greg Egan.
≤0.8440935944\le 0.8440935944 [Gib2018]

Known lower bounds

Bound Reference Comments
π4=0.7853981634…\frac{\pi}{4}=0.7853981634\dots Trivial Use unit disk
0.82570.8257 [Elek1994] Use unit disk and equilateral triangle
0.82710.8271 [Elek1994] Also use regular 3j3^j-gons
0.8320.832 [BS2005] Rigorous computer-aided search using a circle, equilateral triangle, and regular pentagon
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Lebesgue's convex universal cover constant ($C_{13b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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