Moser's worm problem

For example, a circular disk of radius 1/2 can accommodate any plane curve of length 1 by placing the midpoint of the curve at the center of the disk. Another possible solution has the shape of a rhombus with vertex angles of 60° and 120° and with a long diagonal of unit length.[1] However, these are not optimal solutions; other shapes are known that solve the problem with smaller areas.

It is not completely trivial that a minimum-area cover exists. An alternative possibility would be that there is some minimal area that can be approached but not actually attained. However, there does exist a smallest convex cover. Its existence follows from the Blaschke selection theorem.[2]

It is also not trivial to determine whether a given shape forms a cover. Gerriets & Poole (1974) conjectured that a shape accommodates every unit-length curve if and only if it accommodates every unit-length polygonal chain with three segments, a more easily tested condition, but Panraksa, Wetzel & Wichiramala (2007) showed that no finite bound on the number of segments in a polychain would suffice for this test.

The problem remains open, but over a sequence of papers researchers have tightened the gap between the known lower and upper bounds. In particular, Norwood & Poole (2003) constructed a (nonconvex) universal cover and showed that the minimum shape has area at most 0.260437; Gerriets & Poole (1974) and Norwood, Poole & Laidacker (1992) gave weaker upper bounds. In the convex case, Wang (2006) improved an upper bound to 0.270911861. Khandhawit, Pagonakis & Sriswasdi (2013) used a min-max strategy for area of a convex set containing a segment, a triangle and a rectangle to show a lower bound of 0.232239 for a convex cover.

In the 1970s, John Wetzel conjectured that a 30° circular sector of unit radius is a cover with area ${\displaystyle \pi /12\approx 0.2618}$. Two proofs of the conjecture were independently claimed by Movshovich & Wetzel (2017) and by Panraksa & Wichiramala (2021). If confirmed, this will reduce the upper bound for the convex cover by about 3%.

• Moving sofa problem, the problem of finding a maximum-area shape that can be rotated and translated through an L-shaped corridor
• Kakeya set, a set of minimal area that can accommodate every unit-length line segment (with translations allowed, but not rotations)
• Lebesgue's universal covering problem, find the smallest convex area that can cover any planar set of unit diameter
• Bellman's lost in a forest problem, find the shortest path to escape from a forest of known size and shape.

• Gerriets, John; Poole, George (1974), "Convex regions which cover arcs of constant length", The American Mathematical Monthly, 81 (1): 36–41, doi:10.2307/2318909, JSTOR 2318909, MR 0333991.
• Khandhawit, Tirasan; Pagonakis, Dimitrios; Sriswasdi, Sira (2013), "Lower Bound for Convex Hull Area and Universal Cover Problems", International Journal of Computational Geometry & Applications, 23 (3): 197–212, arXiv:1101.5638, doi:10.1142/S0218195913500076, MR 3158583, S2CID 207132316.
• Norwood, Rick; Poole, George (2003), "An improved upper bound for Leo Moser's worm problem", Discrete and Computational Geometry, 29 (3): 409–417, doi:10.1007/s00454-002-0774-3, MR 1961007.
• Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete and Computational Geometry, 7 (2): 153–162, doi:10.1007/BF02187832, MR 1139077.
• Panraksa, Chatchawan; Wetzel, John E.; Wichiramala, Wacharin (2007), "Covering n-segment unit arcs is not sufficient", Discrete and Computational Geometry, 37 (2): 297–299, doi:10.1007/s00454-006-1258-7, MR 2295060.
• Wang, Wei (2006), "An improved upper bound for the worm problem", Acta Mathematica Sinica, 49 (4): 835–846, MR 2264090.
• Panraksa, Chatchawan; Wichiramala, Wacharin (2021), "Wetzel's sector covers unit arcs", Periodica Mathematica Hungarica, 82 (2): 213–222, arXiv:1907.07351, doi:10.1007/s10998-020-00354-x, S2CID 225397486.
• Movshovich, Yevgenya; Wetzel, John (2017), "Drapeable unit arcs fit in the unit 30° sector", Advances in Geometry, 17 (4): 497–506, doi:10.1515/advgeom-2017-0011, S2CID 125746596.