In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width. In the form that every curve of a given width has area at least as large as the Reuleaux triangle, it is also known as the Blaschke–Lebesgue inequality. It is named after Wilhelm Blaschke and Henri Lebesgue, who published it separately in the early 20th century.
The width of a convex set
K
K
w
1 | |
2 |
(\pi-\sqrt3)w2 ≈ 0.70477w2.
w
The Blaschke–Lebesgue theorem was published independently in 1914 by Henri Lebesgue and in 1915 by Wilhelm Blaschke. Since their work, several other proofs have been published.
The same theorem is also true in the hyperbolic plane. 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.
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. For a ship with
n
O(loglogn)
By the isoperimetric inequality, the curve of constant width in the Euclidean plane with the largest area is a circle. The perimeter of a curve of constant width
w
\piw
It is unknown which surfaces of constant width in three-dimensional space have the minimum volume. 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, but this remains unproven.