Lebesgue's universal covering problem

For other uses of "universal cover" or "universal covering", see Covering space § Universal covers.

An equilateral triangle of diameter 1 doesn’t fit inside a circle of diameter 1

Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every 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.

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)

Formulation and early research

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 $${\displaystyle 2-{\frac {2}{\sqrt {3}}}\approx 0.84529946.}$$

The shape outlined in black is Pál's solution to Lebesgue's universal covering problem. Within it, planar shapes with diameter one have been included: a circle (in blue), a Reuleaux triangle (in red) and a square (in green).

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 ${\displaystyle a\leq 0.844137708436}$.

Current bounds

After a sequence of improvements to Sprague's solution, each removing small corners from the solution,[3][4] a 2018 preprint of Philip Gibbs claimed the best upper bound known, a further reduction to area 0.8440935944.[5][6]

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, proving that the area of an optimal cover is at least 0.832.[7]

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)
• Blaschke selection theorem, which can be used to prove that Lebesgue's universal covering problem has a solution.

References

1. Pál, J. (1920). "'Über ein elementares Variationsproblem". Danske Mat.-Fys. Meddelelser III. 2.
2. Sprague, R. (1936). "Über ein elementares Variationsproblem". Matematiska Tidsskrift Ser. B: 96–99. JSTOR 24530328.
3. Hansen, H. C. (1992). "Small universal covers for sets of unit diameter". Geometriae Dedicata. 42 (2): 205–213. doi:10.1007/BF00147549. MR 1163713. S2CID 122081393.
4. Baez, John C.; Bagdasaryan, Karine; Gibbs, Philip (2015). "The Lebesgue universal covering problem". Journal of Computational Geometry. 6: 288–299. arXiv:1502.01251. doi:10.20382/jocg.v6i1a12. MR 3400942. S2CID 20752239.
5. Gibbs, Philip (23 October 2018). "An upper bound for Lebesgue's covering problem". arXiv:1810.10089 [math.MG].
6. "Amateur mathematician finds smallest universal cover". Quanta Magazine. Archived from the original on 2019-01-14. Retrieved 2018-11-16.
7. Brass, Peter; Sharifi, Mehrbod (2005). "A lower bound for Lebesgue's universal cover problem". International Journal of Computational Geometry and Applications. 15 (5): 537–544. doi:10.1142/S0218195905001828. MR 2176049.