Moser’s convex worm cover constant
Description of constant
$C_{13a}$ is the infimal area of a convex domain $\Omega$ that can contain a rigid motion (translation + rotation; no reflections) of every planar arc (curve, or “worm”) of length $1$.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $\frac{\pi}{4}=0.7853981633\dots$ | Trivial | disk of radius $1/2$ |
| $\frac{\pi}{8}=0.3926990816\dots$ | Meir [Wet1973] | semicircle of radius $1/2$ |
| $0.34501$ | [Wet1973] | |
| $0.3214$ | [G1972] | |
| $0.28610$ | [GP1974] | |
| $0.27524$ | [NPL1992] | |
| $0.270911861$ | [Wan2006] | |
| $\frac{\pi}{12}=0.2617993878\dots$ | [PW2021] | 30° circular sector of radius 1 (“Wetzel’s sector”). Builds upon [MW2017] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0.2194$ | [Wet1973] | Uses “broadworms” (building on work of Schaer on the broadest curve of unit length) |
| $0.227498$ | [KS2009] | Uses “V”-worms (equilateral-triangle hull) and “U”-worms (square hull) |
| $0.232239$ | [KPS2013] | Min–max estimate for convex hull area of certain forced configurations |
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:
- $\operatorname{Area}(\Omega)\le 0.2604370$ [NP2003].
- $\operatorname{Area}(\Omega)\le 0.26007$ [PW2018].
However, the constant is still positive in this case [Mar1979], [FO2018].
- It is not sufficient to test covering of polygonal paths [PWW2007].
- Wikipedia page for this problem
- See also: Lebesgue’s universal covering problem.
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 Unpublished, 1972.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. 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 $\mathbb{R}^q$. Mathematika 26 (1979), 1–12.
- [MW2017] Movshovich, Yevgenya; Wetzel, John E. Drapeable unit arcs fit in the unit $30^\circ$ 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 $n$-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.
Contribution notes
ChatGPT DeepResearch was used to prepare an initial version of this page.