|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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>
| | Andrew Simcox is the name his parents gave him and he totally loves this name. To perform lacross is one of the things she loves most. Her family members lives in Alaska but her spouse desires them to transfer. Office supervising is what she does for a living.<br><br>Feel free to visit my web blog: [http://brazil.amor-amore.com/irboothe free psychic reading] |
| | |
| 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]]
| |
Latest revision as of 03:48, 7 November 2014
Andrew Simcox is the name his parents gave him and he totally loves this name. To perform lacross is one of the things she loves most. Her family members lives in Alaska but her spouse desires them to transfer. Office supervising is what she does for a living.
Feel free to visit my web blog: free psychic reading