[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/blaschke-lebesgue-theorem-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/blaschke-lebesgue-theorem-wikipedia\/","headline":"Blaschke\u2013Lebesgue theorem – Wikipedia","name":"Blaschke\u2013Lebesgue theorem – Wikipedia","description":"From Wikipedia, the free encyclopedia Plane geometry theorem on least area of all curves of given constant width In plane","datePublished":"2019-08-05","dateModified":"2019-08-05","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/eb\/Reuleaux_supporting_lines.svg\/220px-Reuleaux_supporting_lines.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/eb\/Reuleaux_supporting_lines.svg\/220px-Reuleaux_supporting_lines.svg.png","height":"165","width":"220"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/blaschke-lebesgue-theorem-wikipedia\/","about":["Wiki"],"wordCount":5026,"articleBody":"From Wikipedia, the free encyclopediaPlane geometry theorem on least area of all curves of given constant width  In plane geometry the Blaschke\u2013Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width.[1] 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\u2013Lebesgue inequality.[2] It is named after Wilhelm Blaschke and Henri Lebesgue, who published it separately in the early 20th century.Table of ContentsStatement[edit]History[edit]In other planes[edit]Application[edit]Related problems[edit]References[edit]Statement[edit]The width of a convex set K{displaystyle K} 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 tangent lines to K{displaystyle K}, on opposite sides. 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. These curves include both the circle and the Reuleaux triangle, a curved triangle formed from arcs of three equal-radius circles each centered at a crossing point of the other two circles. The area enclosed by a Reuleaux triangle with width w{displaystyle w} is12(\u03c0\u22123)w2\u22480.70477w2.{displaystyle {frac {1}{2}}(pi -{sqrt {3}})w^{2}approx 0.70477w^{2}.}The Blaschke\u2013Lebesgue theorem states that this is the unique minimum possible area of a curve of constant width, and the Blaschke\u2013Lebesgue inequality states that every convex set of width w{displaystyle w} has area at least this large, with equality only when the set is bounded by a Reuleaux triangle.[1]History[edit]The Blaschke\u2013Lebesgue 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[edit]The same theorem is also true in the hyperbolic plane.[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]Application[edit]The Blaschke\u2013Lebesgue 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{displaystyle n} grid points, it is possible to bound the number of missed shots by O(log\u2061log\u2061n){displaystyle O(log log n)}.[14]Related problems[edit]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 w{displaystyle w} is \u03c0w{displaystyle pi w}, 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[edit]^ a b c Gruber, Peter M. (1983), Convexity and its Applications, Birkh\u00e4user, p.\u00a067, ISBN\u00a0978-3-7643-1384-5^ Martini, Horst; Montejano, Luis; Oliveros, D\u00e9borah (2019), Bodies of Constant Width: An introduction to convex geometry with applications, Birkh\u00e4user\/Springer, Cham, p.\u00a0336, doi:10.1007\/978-3-030-03868-7, ISBN\u00a0978-3-030-03866-3, MR\u00a03930585^ Lebesgue, Henri (1914), “Sur le probl\u00e8me des isop\u00e9rim\u00e8tres et sur les domaines de largeur constante”, Bulletin de la Soci\u00e9t\u00e9 Math\u00e9matique de France, 7: 72\u201376^ Blaschke, Wilhelm (1915), “Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts”, Mathematische Annalen, 76 (4): 504\u2013513, doi:10.1007\/BF01458221, MR\u00a01511839^ Fujiwara, Matsusabur\u00f4 (1927), “Analytic proof of Blaschke’s theorem on the curve of constant breadth with minimum area”, Proceedings of the Imperial Academy, 3 (6): 307\u2013309, MR\u00a01568234; Fujiwara, Matsusaburo (1931), “Analytic proof of Blaschke’s theorem on the curve of constant breadth, II”, Proceedings of the Imperial Academy, 7 (8): 300\u2013302, MR\u00a01568319^ Mayer, Anton E. (1935), “Der Inhalt der Gleichdicke”, Mathematische Annalen, 110 (1): 97\u2013127, doi:10.1007\/BF01448020, MR\u00a01512931^ Eggleston, H. G. (1952), “A proof of Blaschke’s theorem on the Reuleaux triangle”, Quarterly Journal of Mathematics, Second Series, 3: 296\u2013297, doi:10.1093\/qmath\/3.1.296, MR\u00a00051543^ Ghandehari, Mostafa (1996), “An optimal control formulation of the Blaschke-Lebesgue theorem”, Journal of Mathematical Analysis and Applications, 200 (2): 322\u2013331, doi:10.1006\/jmaa.1996.0208, MR\u00a01391153^ Harrell, Evans M. II (2002), “A direct proof of a theorem of Blaschke and Lebesgue”, The Journal of Geometric Analysis, 12 (1): 81\u201388, doi:10.1007\/BF02930861, MR\u00a01881292^ Malagoli, Federica (2009), “An optimal control theory approach to the Blaschke\u2013Lebesgue theorem”, Journal of Convex Analysis, 16 (2): 391\u2013407, MR\u00a02559951^ Ara\u00fajo, Paulo Ventura (1997), “Minimum area of a set of constant width in the hyperbolic plane”, Geometriae Dedicata, 64 (1): 41\u201353, doi:10.1023\/A:1004920201363, MR\u00a01432533^ Ohmann, D. (1952), “Extremalprobleme f\u00fcr konvexe Bereiche der euklidischen Ebene”, Mathematische Zeitschrift, 55: 346\u2013352, doi:10.1007\/BF01181132, MR\u00a00048831^ Chakerian, G. D. (1966), “Sets of constant width”, Pacific Journal of Mathematics, 19: 13\u201321, MR\u00a00205152^ Crombez, Lo\u00efc; da Fonseca, Guilherme D.; Gerard, Yan (2020), “Efficient algorithms for Battleship”,  in Farach-Colton, Martin; Prencipe, Giuseppe; Uehara, Ryuhei (eds.), 10th International Conference on Fun with Algorithms (FUN 2021), Leibniz International Proceedings in Informatics (LIPIcs), vol.\u00a0157, Dagstuhl, Germany: Schloss Dagstuhl\u2013Leibniz-Zentrum f\u00fcr Informatik, pp.\u00a011:1\u201311:15, doi:10.4230\/LIPIcs.FUN.2021.11, ISBN\u00a0978-3-95977-145-0^ Barbier, E. (1860), “Note sur le probl\u00e8me de l’aiguille et le jeu du joint couvert” (PDF), Journal de math\u00e9matiques pures et appliqu\u00e9es, 2e s\u00e9rie (in French), 5: 273\u2013286. See in particular pp. 283\u2013285.^ Bonnesen, Tommy; Fenchel, Werner (1934), Theorie der konvexen K\u00f6rper, Springer-Verlag, pp.\u00a0127\u2013139^ Anciaux, Henri; Guilfoyle, Brendan (2011), “On the three-dimensional Blaschke\u2013Lebesgue problem”, Proceedings of the American Mathematical Society, 139 (5): 1831\u20131839, doi:10.1090\/S0002-9939-2010-10588-9, MR\u00a02763770  "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/blaschke-lebesgue-theorem-wikipedia\/#breadcrumbitem","name":"Blaschke\u2013Lebesgue theorem – Wikipedia"}}]}]