Lebesgue universal covering problem

Lebesgue's universal covering problem asks for the convex shape of smallest area that can cover¹ any planar set of diameter one, that is, the least upper bound of the distances between all pairs of points in the set.


The problem was posed by Henri Lebesgue in a letter to Gyula Pál in 1914. Within this file there is Pál's universal cover along with some sets with unit diameter. Recently, new universal covers with smaller area have been discovered, and the cover of minimal area is still unknown. For more information, see: https://www.quantamagazine.org/amateur-mathematician-finds-smallest-universal-cover-20181115


Notes:
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(1): A shape covers a set if the set may be rotated, translated or reflected to fit inside the shape.