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Lebesgue's universal covering problem is an unsolved problem in
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. This reduced the upper bound on the area to .
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 f ...
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 byHenri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
in a letter to Gyula Pál 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'' has ...
with an inscribed circle of diameter one and removing two corners from the hexagon to give a cover of area
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. 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
In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area that can be maneuvered through an L-shaped planar region ...
, 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
{{reflist Discrete geometry Unsolved problems in geometry