Moser's worm problem

Unsolved problem in mathematics: What is the minimum area of a shape that can cover every unit-length curve? (more unsolved problems in mathematics)

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.

Examples

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 degrees (π/3 and 2π/3 radians) 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.

Solution properties

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.^[2]

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.

Known bounds

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 (2011) 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.

See also

• 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

Notes

1. Gerriets & Poole (1974).
2. Norwood, Poole & Laidacker (1992) attribute this observation to an unpublished manuscript of Laidacker and Poole, dated 1986.

References

• 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 (2013), "Lower Bound for Convex Hull Area and Universal Cover Problems", International Journal of Computational Geometry & Applications, 23 (3): 197–212, doi:10.1142/S0218195913500076, MR 3158583.
• 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.