Blaschke–Lebesgue theorem
In
Statement
The width of a convex set in the Euclidean plane is defined as the minimum distance between any two parallel lines that enclose it. The two minimum-distance lines are both necessarily
The Blaschke–Lebesgue theorem states that this is the unique minimum possible area of a curve of constant width, and the Blaschke–Lebesgue inequality states that every convex set of width has area at least this large, with equality only when the set is bounded by a Reuleaux triangle.[1]
History
The Blaschke–Lebesgue theorem was published independently in 1914 by Henri Lebesgue[3] and in 1915 by Wilhelm Blaschke.[4] Since their work, several other proofs have been published.[5][6][7][8][9][10]
In other planes
The same theorem is also true in the
Application
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 grid points, it is possible to bound the number of missed shots by .[14]
Related problems
By the isoperimetric inequality, the curve of constant width in the Euclidean plane with the largest area is a circle.[1] The perimeter of a curve of constant width is , regardless of its shape; this is Barbier's theorem.[15]
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,[16] but this remains unproven.[17]
References
- ^ ISBN 978-3-7643-1384-5
- MR 3930585
- ^ Lebesgue, Henri (1914), "Sur le problème des isopérimètres et sur les domaines de largeur constante", Bulletin de la Société Mathématique de France, 7: 72–76
- MR 1511839
- MR 1568319
- MR 1512931
- MR 0051543
- MR 1391153
- MR 1881292
- MR 2559951
- MR 1432533
- MR 0048831
- MR 0205152
- ISBN 978-3-95977-145-0
- ^ Barbier, E. (1860), "Note sur le problème de l'aiguille et le jeu du joint couvert" (PDF), Journal de mathématiques pures et appliquées, 2e série (in French), 5: 273–286. See in particular pp. 283–285.
- ^ Bonnesen, Tommy; Fenchel, Werner (1934), Theorie der konvexen Körper, Springer-Verlag, pp. 127–139
- MR 2763770