The least-area convex cover of every unit-length planar curve
Let $\mathcal C$ be the class of compact convex sets $K \subset \mathbb{R}^2$ with the following property: for every continuous rectifiable curve $\gamma\colon [0,1] \to \mathbb{R}^2$ of length one, which may self-intersect and need not be closed, there are a rotation $R \in SO(2)$ and a vector $a \in \mathbb{R}^2$ such that $R\gamma([0,1]) + a \subset K$. Moser asked for the least area of a planar region, not necessarily convex, with this property; this is the convex version of his worm problem. By the Blaschke selection theorem a minimizer over $\mathcal C$ exists (see Norwood, Poole and Laidacker). Khandhawit, Pagonakis and Sriswasdi proved that every $K \in \mathcal C$ has area at least $0.232239$, and Panraksa and Wichiramala proved that the circular sector of radius one and angle $\pi/6$, of area $\pi/12 = 0.2617\ldots$, belongs to $\mathcal C$. A 2026 preprint of Keller, computer-assisted and formalized in Lean, gives the bounds $0.239$ and $0.24633\ldots$ for the least area.
Problem. Determine $\inf_{K \in \mathcal C} \operatorname{Area}(K)$ and characterize the minimizers up to rigid motion.
Source: Moser (1966), for covers that need not be convex; the convex version as in Khandhawit, Pagonakis and Sriswasdi (2013), §1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, Moser's worm problem, universal covers
References
@misc{moser1966,
author = {Moser, Leo},
title = {Poorly formulated unsolved problems of combinatorial geometry},
howpublished = {Mimeographed list},
year = {1966},
}
@article{khandhawit2013,
author = {Khandhawit, Tirasan and Pagonakis, Dimitrios and Sriswasdi, Sira},
title = {Lower bound for convex hull area and universal cover problems},
journal = {Internat. J. Comput. Geom. Appl.},
volume = {23},
number = {3},
year = {2013},
pages = {197--212},
doi = {10.1142/S0218195913500076},
eprint = {1101.5638},
archivePrefix = {arXiv},
}
@article{norwood1992,
author = {Norwood, Rick and Poole, George and Laidacker, Michael},
title = {The worm problem of {L}eo {M}oser},
journal = {Discrete Comput. Geom.},
volume = {7},
number = {2},
year = {1992},
pages = {153--162},
doi = {10.1007/BF02187832},
}
@article{panraksa2021,
author = {Panraksa, Chatchawan and Wichiramala, Wacharin},
title = {Wetzel's sector covers unit arcs},
journal = {Period. Math. Hungar.},
volume = {82},
number = {2},
year = {2021},
pages = {213--222},
doi = {10.1007/s10998-020-00354-x},
eprint = {1907.07351},
archivePrefix = {arXiv},
}
@misc{keller2026,
author = {Keller, Ethan},
title = {Improved bounds for universal convex covers of unit arcs},
year = {2026},
eprint = {2609.21968},
archivePrefix = {arXiv},
}
@misc{aim-convex-worm-cover,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The least-area convex cover of every unit-length planar curve}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/243-convex-worm-cover.md},
note = {Entry 243 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00514, "The least-area convex cover of every unit-length planar curve", clopen.solutions.