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OP-12014OpenGeometry and topology

Moser's convex worm cover constant (C13aC_{13a})

Moser's convex worm cover constant: determine C13aC_{13a} or improve its bounds; the repository records 0.232239 (lower) and 0.2617993878 (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

C13aC_{13a} is the infimal area of a convex domain Ω\Omega that can contain a rigid motion (translation + rotation; no reflections) of every planar arc (curve, or "worm") of length 11.

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Background

  • The Blaschke selection theorem implies that a minimal convex cover exists.
  • If one drops convexity, one can improve the upper bound as follows:
  • Area⁡(Ω)≤0.2604370\operatorname{Area}(\Omega)\le 0.2604370 [NP2003].
  • Area⁡(Ω)≤0.26007\operatorname{Area}(\Omega)\le 0.26007 [PW2018].

However, the constant is still positive in this case [Mar1979], [FO2018].

References (from the repository page)

  • [FO2018] Fässler, Katrin; Orponen, Tuomas. Curve packing and modulus estimates. arXiv:1602.01707. (Published version: Trans. Amer. Math. Soc. 370 (2018).)
  • [G1972] Gerriets, J. An improved solution to Moser’s worm problem. Unpublished, 1972.
  • [GP1974] Gerriets, J.; Poole, G. An improved solution to Moser’s worm problem. American Mathematical Monthly 81 (1974), no. 1, 36–41. DOI: 10.2307/2318213.
  • [KPS2013] Khandhawit, Tirasan; Pagonakis, Dimitrios; Sriswasdi, Sira. Lower bound for convex hull area and universal cover problems. International Journal of Computational Geometry & Applications 23 (2013), no. 3, 197–212. DOI: 10.1142/S0218195913500076. arXiv:1101.5638.
  • [KS2009] Khandhawit, Tirasan; Sriswasdi, Sira. An Improved Lower Bound for Moser's Worm Problem. arXiv:math/0701391 (v2, 2009).
  • [Mar1979] Marstrand, J. M. Packing smooth curves in Rq\mathbb{R}^q. Mathematika 26 (1979), 1–12.
  • [MW2017] Movshovich, Yevgenya; Wetzel, John E. Drapeable unit arcs fit in the unit 30∘30^\circ sector. Advances in Geometry 17 (2017). DOI: 10.1515/advgeom-2017-0011.
  • [NPL1992] Norwood, Rick; Poole, George; Laidacker, Michael. The worm problem of Leo Moser. Discrete & Computational Geometry 7 (1992), 153–162. DOI: 10.1007/BF02187832.
  • [NP2003] Norwood, Rick; Poole, George. An improved upper bound for Leo Moser’s worm problem. Discrete & Computational Geometry 29 (2003), 409–417. DOI: 10.1007/s00454-002-0774-3.
  • [PW2018] Ploymaklam, Nattapol; Wichiramala, Wacharin. A Smaller Cover of the Moser’s Worm Problem. Chiang Mai Journal of Science 45 (2018), no. 6, 2528–2533. (Open-access PDF: https://www.thaiscience.info/Journals/Article/CMJS/10990404.pdf)
  • [PW2021] Panraksa, Chatchawan; Wichiramala, Wacharin. Wetzel’s sector covers unit arcs. Periodica Mathematica Hungarica 82 (2021), 213–222. DOI: 10.1007/s10998-020-00354-x. arXiv:1907.07351.
  • [PWW2007] Panraksa, Chatchawan; Wetzel, John E.; Wichiramala, Wacharin. Covering nn-segment unit arcs is not sufficient. Discrete & Computational Geometry 37 (2007), 297–299. DOI: 10.1007/s00454-006-1258-7.
  • [Wan2006] Wang, Wei. An improved upper bound for the worm problem. Acta Mathematica Sinica (Chinese Series) 49 (2006), no. 4, 835–846. DOI: 10.12386/A2006sxxb0103.
  • [Wet1973] Wetzel, John E. Sectorial covers for curves of constant length. Canadian Mathematical Bulletin 16 (1973), 367–376.
  • [Wet2005] Wetzel, John E. The Classical Worm Problem — A Status Report. Geombinatorics 15 (2005), no. 1, 34–42.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.232239; best known upper bound 0.2617993878. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
π4=0.7853981633…\frac{\pi}{4}=0.7853981633\dots Trivial disk of radius 1/21/2
π8=0.3926990816…\frac{\pi}{8}=0.3926990816\dots Meir [Wet1973] semicircle of radius 1/21/2
0.345010.34501 [Wet1973]
0.32140.3214 [G1972]
0.286100.28610 [GP1974]
0.275240.27524 [NPL1992]
0.2709118610.270911861 [Wan2006]
π12=0.2617993878…\frac{\pi}{12}=0.2617993878\dots [PW2021] 30° circular sector of radius 1 (“Wetzel’s sector”). Builds upon [MW2017]

Known lower bounds

Bound Reference Comments
0.21940.2194 [Wet1973] Uses “broadworms” (building on work of Schaer on the broadest curve of unit length)
0.2274980.227498 [KS2009] Uses “V”-worms (equilateral-triangle hull) and “U”-worms (square hull)
0.2322390.232239 [KPS2013] Min–max estimate for convex hull area of certain forced configurations
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Moser's convex worm cover constant ($C_{13a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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