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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.
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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
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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.
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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.
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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
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in 1914. It was published in a paper by Pál in 1920 along with Pál's analysis. He showed that a cover for all
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with an inscribed circle of diameter one and removing two corners from the hexagon to give a cover of area
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one is also a cover for all sets of diameter one and that a cover can be constructed by taking a regular
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Brass, Peter; Sharifi, Mehrbod (2005). "A lower bound for
Lebesgue's universal cover problem".
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What is the minimum area of a convex shape that can cover every planar set of diameter one?
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Gibbs, Philip (23 October 2018). "An upper bound for
Lebesgue's covering problem".
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An equilateral triangle of diameter 1 doesn’t fit inside a circle of diameter 1
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shape of smallest area that can cover every planar set of diameter one. The
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Hansen, H. C. (1992). "Small universal covers for sets of unit diameter".
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For other uses of "universal cover" or "universal covering", see
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International
Journal of Computational Geometry and Applications
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124:{\displaystyle 2-{\frac {2}{\sqrt {3}}}\approx 0.84529946.}
400:"Amateur mathematician finds smallest universal cover"
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64:(more unsolved problems in mathematics)
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21:Covering space § Universal covers
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35:Lebesgue's universal covering problem
165:{\displaystyle a\leq 0.844137708436}
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213:Blaschke selection theorem
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333:: 288–299.
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468:Categories
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220:References
207:Kakeya set
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256:: 96–99.
157:≤
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80:Gyula Pál
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189:See also
47:diameter
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138:green).
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