Open problem, Geometry, posed 1966

Moser's worm problem

Open

Find a measurable region $K \subset \mathbb{R}^2$ (often additionally required to be convex) of minimum possible area $\mu(K)$ such that every rectifiable planar curve of length $1$ can be placed inside $K$ by a rigid motion (translation and rotation).

Research frontier as of 2026

As of 2026, both the convex and non-convex versions of Moser's worm problem remain open, and the exact shape of the optimal cover is unknown. For convex covers under rigid motions, the minimum area $\mu_{\text{conv}}$ is trapped in the interval $0.232239 \le \mu_{\text{conv}} \le \pi/12 \approx 0.2618$ . When non-convex covers are permitted, Norwood and Poole's construction achieves area $0.260437$ , strictly smaller than the best peer-reviewed convex cover, though it is not even known whether the true infimum for non-convex covers is attained by a region with piecewise-smooth boundary.

Best known results

For convex covers, the smallest peer-reviewed area is $\mu_{\text{conv}} \le \pi/12 \approx 0.261799$ via the $30^\circ$ unit circular sector (Panraksa and Wichiramala, 2021).
For arbitrary measurable covers (allowing non-convexity), the smallest known area is $\mu \le 0.260437$ (Norwood and Poole, 2003).
The best certified lower bound for convex universal covers is $\mu_{\text{conv}} \ge 0.232239$ (Khandhawit, Pagonakis, and Sriswasdi, 2013).

Tools and where they stop

Tool	Achieved	Where it stops
Finite test-set convex hull optimization (Khandhawit–Pagonakis–Sriswasdi)	Establishes $\mu_{\text{conv}} \ge 0.232239$ by minimizing the convex hull area of placements of a few explicit curves (a unit segment, an equilateral triangle, and a rectangle)	Adding more test curves causes the dimension of the configuration space of rigid placements to explode exponentially, making rigorous branch-and-bound verification intractable
Circular sectors and geometric clipping (Wetzel, Norwood–Poole, Wang, Panraksa–Wichiramala)	Yields the best peer-reviewed upper bounds $\pi/12 \approx 0.2618$ (convex) and $0.260437$ (non-convex) by showing every unit curve can be rotated so its support and span fit inside a sector or clipped region	Relies on hand-crafted families of regions and sufficient conditions based on curve diameter/width rather than a variational characterization of an optimal boundary

Open questions

What is the exact minimum area $\mu_{\text{conv}}$ of a convex universal cover for unit-length planar curves?
Does there exist a minimum-area non-convex measurable cover, and is its area strictly smaller than $\mu_{\text{conv}}$ ?

References

Rick Norwood, George Poole (2003). An improved upper bound for Leo Moser's worm problem · DOI:10.1007/s00454-002-0774-3
Chatchawan Panraksa, Wacharin Wichiramala (2021). Wetzel's sector covers unit arcs · DOI:10.1007/s10998-020-00354-x · arXiv:1907.07351
Tirasan Khandhawit, Dimitrios Pagonakis, Sira Sriswasdi (2013). Lower bound for convex hull area and universal cover problems · DOI:10.1142/S0218195913500076 · arXiv:1101.5638