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 every planar set of diameter one.

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.

After a sequence of improvements to Sprague's solution, each removing small corners from the solution,[3][4] a 2018 preprint of Philip Gibbs claimed the best upper bound known, a further reduction to area 0.8440935944.

[5][6] 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.

An equilateral triangle of diameter 1 doesn’t fit inside a circle of diameter 1
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 green).