Lebesgue'S Universal Covering Problem

(Discrete Geometry)


Discrete Geometry
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What is Lebesgue's universal covering problem?

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. 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 .

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discrete geometryunsolved problems in mathematic

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Sources: DBpedia
 — Date merged: 11/6/2021, 1:32:49 PM
 — Date scraped: 5/20/2021, 5:52:36 PM