Knaster–Kuratowski–Mazurkiewicz lemma
- Knaster–Kuratowski–Mazurkiewicz lemma
The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz in:
B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929) 132–137.
The KKM lemma can be proved from Sperner's lemma and can be used to prove (in fact is equivalent to) the Brouwer fixed point theorem.
KKM Lemma. Suppose that a simplex is covered by the closed sets for and that for all the face of S that is spanned by for is covered by for then all the have a common intersection point.
The two dimensional case may serve as an illustration. In this case the simplex is a triangle, whose vertexes we can label 1, 2 and 3. We are given three closed sets which collectively cover the triangle; also we are told that covers vertex 1, covers vertex 2, covers vertex 3, and that the edge 12 (from vertex 1 to vertex 2) is covered by the sets and , the edge 23 is covered by the sets and , the edge 31 is covered by the sets and . The KKM lemma states that the sets have at least one point in common.
See the proof of KKM Lemma in [http://planetmath.org/encyclopedia/KKMLemma.html Planet Math] .
Wikimedia Foundation.
2010.
Look at other dictionaries:
Lemme de Knaster–Kuratowski–Mazurkiewicz — En mathématiques, et plus précisément en topologie algébrique, le lemme de Knaster–Kuratowski–Mazurkiewicz, ou lemme KKM, est un résultat de point fixe publié en 1929 par Bronisław Knaster (en), Kazimierz Kuratowski et Stefan Mazurkiewicz[1] … Wikipédia en Français
List of mathematics articles (K) — NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) … Wikipedia
List of lemmas — This following is a list of lemmas (or, lemmata , i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. 0 to 9 *0/1 Sorting Lemma ( comparison… … Wikipedia
KKM — is a three letter abbreviation that may refer to:* Knaster Kuratowski Mazurkiewicz lemma, in mathematics * Kyo Kara Maoh!, a series of Japanese light novels * Lop Buri Airport, Thailand … Wikipedia
Lemme de Sperner — Ne pas confondre avec le théorème de Sperner sur les familles d ensembles. En mathématiques, le lemme de Sperner, dû à Emanuel Sperner[1], est un analogue combinatoire du théorème du point fixe de Brouwer. Le lemme de Sperner affirme que… … Wikipédia en Français
