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The '''Knaster–Kuratowski–Mazurkiewicz lemma''' is a basic result in mathematical [[fixed-point theorem|fixed-point theory]] published in 1929 by [[Bronisław Knaster|Knaster]], [[Kazimierz Kuratowski|Kuratowski]] and [[Stefan Mazurkiewicz|Mazurkiewicz]].<ref>{{citation|first1=B.|last1=Knaster|author1-link=Bronisław Knaster|first2=C.|last2=Kuratowski|author2-link=Kazimierz Kuratowski|first3=S.|last3=Mazurkiewicz|author3-link=Stefan Mazurkiewicz|title=Ein Beweis des Fixpunktsatzes für ''n''-dimensionale Simplexe|journal=[[Fundamenta Mathematicae]]|volume=14|issue=1|year=1929|pages=132–137|language=German|url=https://eudml.org/doc/212127}}.</ref> | |||
The KKM lemma can be proved from [[Sperner's lemma]] and can be used to prove the [[Brouwer fixed-point theorem]]. | |||
==Statement of the lemma== | |||
KKM Lemma. Suppose that a [[simplex]] <math>\Delta_m</math> is covered by the [[closed set]]s <math>C_i</math> for <math>i \in I=\{1,\dots,m\}</math> and that for all <math>I_k \subset I</math> the face of <math>\Delta_m</math> that is spanned by <math>e_i</math> for <math>i \in I_k</math> is covered by <math>C_i </math> for <math>i \in I_k</math> then all the <math>C_i</math> have a common intersection point. | |||
==Example== | |||
The two-dimensional case may serve as an illustration. In this case the simplex <math>\Delta_3</math> is a triangle, whose vertices we can label 1, 2 and 3. We are given three closed sets <math>C_1,C_2,C_3</math> which collectively cover the triangle; also we are told that <math>C_1</math> covers vertex 1, <math>C_2</math> covers vertex 2, <math>C_3</math> covers vertex 3, and that the edge 12 (from vertex 1 to vertex 2) is covered by the sets <math>C_1</math> and <math>C_2</math>, the edge 23 is covered by the sets <math>C_2</math> and <math>C_3</math>, the edge 31 is covered by the sets <math>C_3</math> and <math>C_1</math>. The KKM lemma states that the sets <math>C_1, C_2, C_3</math> have at least one point in common. | |||
==References== | |||
{{reflist}} | |||
==External links== | |||
*See the proof of KKM Lemma in [http://planetmath.org/encyclopedia/KKMLemma.html Planet Math]. | |||
{{DEFAULTSORT:Knaster-Kuratowski-Mazurkiewicz lemma}} | |||
[[Category:Fixed points (mathematics)]] | |||
[[Category:Lemmas]] | |||
Revision as of 04:42, 19 January 2014
The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz.[1]
The KKM lemma can be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem.
Statement of the lemma
KKM Lemma. Suppose that a simplex is covered by the closed sets for and that for all the face of that is spanned by for is covered by for then all the have a common intersection point.
Example
The two-dimensional case may serve as an illustration. In this case the simplex is a triangle, whose vertices 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.
References
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External links
- See the proof of KKM Lemma in Planet Math.
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