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Level B · Reproducible Geometry P-constant-13a-moser-s-convex-worm-cover-constant

Moser's convex worm cover constant

C_13a is the infimal area of a convex domain Ω that can contain a rigid motion (translation + rotation; no reflections) of every planar arc (curve, or "worm") of length 1.

Current state (tracked automatically)

  • Best known bounds for constant 13a: 0.239 ≤ C ≤ 0.2617993878. Upper bound ([PW2021], 30° circular sector of radius 1 (“Wetzel’s sector”)); lower bound ([K2026], Theorem 1.1), not yet independently verified.
  • The best independently verified lower bound is 0.232239 ([KPS2013], Min–max estimate for convex hull area of certain forced configurations).
All tracked facts and sources
FactValueSource
Best lower bound with an explicit value (constant 13a)0.239*
[K2026] — Theorem 1.1.
Tao's optimization constants
Best lower bound with an explicit value, independently verified (constant 13a)0.232239
[KPS2013] — Min–max estimate for convex hull area of certain forced configurations
Tao's optimization constants
Best upper bound with an explicit value (constant 13a)0.2617993878
[PW2021] — 30° circular sector of radius 1 (“Wetzel’s sector”).
Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.

From the catalogue. Imported from Terence Tao and contributors (optimizationproblems repository) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-constant-13a-moser-s-convex-worm-cover-constant,
  title        = {Moser's convex worm cover constant},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-13a-moser-s-convex-worm-cover-constant}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board
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Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

or , ,

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

Description of constant

is the infimal area of a convex domain that can contain a rigid motion (translation + rotation; no reflections) of every planar arc (curve, or "worm") of length .

Known upper bounds

BoundReferenceComments
Trivialdisk of radius
Meir [Wet1973]semicircle of radius
[Wet1973]
[G1972]
[GP1974]
[NPL1992]
[Wan2006]
[PW2021]30° circular sector of radius 1 (“Wetzel’s sector”). Builds upon [MW2017]

Known lower bounds

BoundReferenceComments
[Wet1973]Uses “broadworms” (building on work of Schaer on the broadest curve of unit length)
[KS2009]Uses “V”-worms (equilateral-triangle hull) and “U”-worms (square hull)
[KPS2013]Min–max estimate for convex hull area of certain forced configurations
[K2026]Theorem 1.1. Four polygonal unit arcs and a finite subdivision certificate. The paper permits reflections; its lower bound therefore also holds for this page's rotation-and-translation convention. Formalization is reported by the author, without an independent replay recorded here.

Additional comments and links

  • The Blaschke selection theorem implies that a minimal convex cover exists.
  • If one drops convexity, one can improve the upper bound as follows:
  • [NP2003].
  • [PW2018].

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

References

  • [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. Convex regions which cover arcs of constant length. American Mathematical Monthly 81 (1974), no. 1, 36–41. DOI: 10.2307/2318909.
  • [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 . Mathematika 26 (1979), 1–12.
  • [MW2017] Movshovich, Yevgenya; Wetzel, John E. Drapeable unit arcs fit in the unit 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 -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.
  • [K2026] Keller, Ethan. Improved bounds for universal convex covers of unit arcs. arXiv:2609.21968v1, 18 September 2026.

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 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.