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Lebesgue's universal covering problem is an unsolved problem in
In 1936,
Lebesgue's universal covering problem is an unsolved problem in geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
that asks for the convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
shape of smallest area that can cover every planar set of diameter one. The diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
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.
Formulation and early research
The problem was posed by Henri Lebesgue in a letter toGyula Pál
Gyula Pál (27 June 1881 – 6 September 1946) was a noted Hungarian- Danish mathematician. He is known for his work on Jordan curves both in plane and space, and on the Kakeya problem
In mathematics, a Kakeya set, or Besicovitch set, is a s ...
in 1914. It was published in a paper by Pál in 1920 along with Pál's analysis. 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
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A ''regular hexagon'' h ...
with an inscribed circle of diameter one and removing two corners from the hexagon to give a cover of area
In 1936, Roland Sprague
Roland Percival Sprague (11 July 1894, Unterliederbach – 1 August 1967) was a German mathematician, known for the Sprague–Grundy theorem and for being the first mathematician to find a perfect squared square.
Biography
With two mathematicia ...
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. This reduced the upper bound on the area to .
Current bounds
After a sequence of improvements to Sprague's solution, each removing small corners from the solution, a 2018 preprint of Philip Gibbs claimed the best upper bound known, a further reduction to area 0.8440935944. 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.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
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional ...
, a set of minimal area that can accommodate every unit-length line segment (with translations allowed, but not rotations)
*Blaschke selection theorem
The Blaschke selection theorem is a result in topology and convex geometry about sequences of convex sets. Specifically, given a sequence \ of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence \ and a co ...
, which can be used to prove that Lebesgue's universal covering problem has a solution.
References
{{reflist Discrete geometry Unsolved problems in geometry