Lebesgue's universal covering problem
 | Unsolved problem in mathematics: What is the minimum area of a convex shape that can cover every planar set of diameter one? (more unsolved problems in mathematics) |
Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover any planar set of diameter one. The diameter of a set by definition is the least upper bound of the distances between all pairs of points in the set. A shape covers a set if it contains a congruent subset. In other words the set may be rotated, translated or reflected to fit inside the shape.
The problem was posed by Henri Lebesgue in a letter to Gyula Pál in 1914. It was published in a paper by Pál in 1920 along with Pál's analysis.[1] He showed that a cover for all curves of constant width one is also a cover for all sets of diameter one and that a cover can be constructed by taking a regular hexagon with an inscribed circle of diameter one and removing two corners from the hexagon to give a cover of area .
Known bounds
In 1936 Roland Sprague showed that a part of Pál's cover could be removed near one of the other corners while still retaining its property as a cover.[2] This reduced the upper bound on the area to . In 1992 Hansen showed that two more very small regions of Sprague's solution could be removed bringing the upper bound down to . Hansen's construction was the first to make use of the freedom to use reflections.[3] In 2015 John Baez, Karine Bagdasaryan and Philip Gibbs showed that if the corners removed in Pál's cover are cut off at a different angle then it is possible to reduce the area further giving the current best upper bound of .[4]
The best known lower bound for the area was provided by Peter Brass and Mehrbod Sharifi using a combination of three shapes in optimal alignment giving .[5]
See also
- Moser's worm problem, what is the minimum area of a shape that can cover every unit-length curve?
- 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)
References
- ↑ Pál, J. (1920). "'Über ein elementares Variationsproblem". Danske Mat.-Fys. Meddelelser III. 2.
- ↑ Sprague, R. (1936). "Über ein elementares Variationsproblem". Matematiska Tidsskrift Ser. B: 96–99.
- ↑ Hansen, H. C. (1992). "Small universal covers for sets of unit diameter". Geom. Ded. 42: 205–213.
- ↑ Baez, J. C.; Bagdasaryan, K.; Gibbs, P. (2015). "The Lebesgue Universal Covering Problem". Journal of Computational Geometry. 6: 288–299.
- ↑ Brass, P.; Sharifi, M. (2005). "A lower Bound for Lebesgue's Universal Covering Problem". International Journal of Computational Geometry. 15 (5): 537–544.