Moser's worm problem

Short description: Unsolved geometry problem about planar regions

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 (also known as mother worm's blanket 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° 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.

Solution properties

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.

It is also not trivial to determine whether a given shape forms a cover. (Gerriets Poole) 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) 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) constructed a (nonconvex) universal cover and showed that the minimum shape has area at most 0.260437; (Gerriets Poole) and (Norwood Poole) gave weaker upper bounds. In the convex case, (Wang 2006) improved an upper bound to 0.270911861. (Khandhawit Pagonakis) 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 [math]\displaystyle{ \pi/12 \approx 0.2618 }[/math]. Two proofs of the conjecture were independently claimed by (Movshovich Wetzel) and by (Panraksa Wichiramala). If confirmed, this will reduce the upper bound for the convex cover by about 3%.

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
• Bellman's lost in a forest problem, find the shortest path to escape from a forest of known size and shape.

References

• 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 .
• 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 .
• "An improved upper bound for Leo Moser's worm problem", Discrete and Computational Geometry 29 (3): 409–417, 2003, doi:10.1007/s00454-002-0774-3 .
• "The worm problem of Leo Moser", Discrete and Computational Geometry 7 (2): 153–162, 1992, doi:10.1007/BF02187832 .
• 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 .
• Wang, Wei (2006), "An improved upper bound for the worm problem", Acta Mathematica Sinica 49 (4): 835–846 .
• Panraksa, Chatchawan; Wichiramala, Wacharin (2021), "Wetzel's sector covers unit arcs", Periodica Mathematica Hungarica 82 (2): 213–222, doi:10.1007/s10998-020-00354-x, https://link.springer.com/article/10.1007/s10998-020-00354-x .
• 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, https://www.degruyter.com/view/journals/advg/17/4/article-p497.xml .