Blaschke–Lebesgue theorem

In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width.

[2] It is named after Wilhelm Blaschke and Henri Lebesgue, who published it separately in the early 20th century.

A curve of constant width is the boundary of a convex set with the property that, for every direction of parallel lines, the two tangent lines with that direction that are tangent to opposite sides of the curve are at a distance equal to the width.

[11] For any convex distance function on the plane (a distance defined as the norm of the vector difference of points, for any norm), an analogous theorem holds true, according to which the minimum-area curve of constant width is an intersection of three metric disks, each centered on a boundary point of the other two.

[12][13] The Blaschke–Lebesgue theorem has been used to provide an efficient strategy for generalizations of the game of Battleship, in which one player has a ship formed by intersecting the integer grid with a convex set and the other player, after having found one point on this ship, is aiming to determine its location using the fewest possible missed shots.

[14] By the isoperimetric inequality, the curve of constant width in the Euclidean plane with the largest area is a circle.

Bonnesen and Fenchel conjectured in 1934 that the minimizers are the two Meissner bodies obtained by rounding some of the edges of a Reuleaux tetrahedron,[16] but this remains unproven.

In 2011 Anciaux and Guilfoyle [17] proved that the minimizer consists of pieces of spheres and tubes over curves, which is true for the Meissner bodies, thus supporting the conjecture of Bonnesen and Fenchel.