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 S^m is covered by the closed sets C_i for i in I={1,dots,m} and that for all I_k subset I the face of S that is spanned by e_i for i in I_k is covered by C_i for i in I_k then all the C_i have a common intersection point.

The two dimensional case may serve as an illustration. In this case the simplex S^3 is a triangle, whose vertexes we can label 1, 2 and 3. We are given three closed sets C_1,C_2,C_3 which collectively cover the triangle; also we are told that C_1 covers vertex 1, C_2 covers vertex 2, C_3 covers vertex 3, and that the edge 12 (from vertex 1 to vertex 2) is covered by the sets C_1 and C_2, the edge 23 is covered by the sets C_2 and C_3, the edge 31 is covered by the sets C_3 and C_1. The KKM lemma states that the sets C_1, C_2, C_3 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.

Игры ⚽ Поможем написать курсовую

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

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”