Moser's worm problem
What is the minimum area of a shape that can cover every unit-length curve?
Moser's worm problem is an unsolved problem in geometry formulated by the Austrian-Canadian mathematician Leo Moser in 1966. The problem asks for the region of smallest area that can accommodate every plane curve of length 1. Here "accommodate" means that the curve may be rotated and translated to fit inside the region. In some variations of the problem, the region is restricted to be convex.
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 π/3 and 2π/3 and with a long diagonal of unit length. However, these are not optimal solutions; other shapes are known that solve the problem with smaller areas.
It is not completely trivial that a solution exists – an alternative possibility would be that there is some minimal area that can be approached but not actually attained. However, in the convex case, the existence of a solution follows from the Blaschke selection theorem.
It is also not trivial to determine whether a given shape forms a solution. 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) showed that the minimum shape has area at most 0.260437; Gerriets & Poole (1974), Norwood, Poole & Laidacker (1992), and Wang (2006) give weaker upper bounds. Khandhawit, Pagonakis & Sriswasdi (2011) performed computer searches that calculated a lower bound of 0.232239 on the optimal area.
- 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)
- Gerriets, John; Poole, George (1974), "Convex regions which cover arcs of constant length", The American Mathematical Monthly 81: 36–41, doi:10.2307/2318909, MR 0333991.
- Khandhawit, Tirasan; Pagonakis, Dimitrios; Sriswasdi, Sira (2011), Lower Bound for Convex Hull Area and Universal Cover Problems, arXiv:1101.5638.
- 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.