Knaster–Kuratowski–Mazurkiewicz lemma

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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[edit]

KKM Lemma. Suppose that a simplex \Delta_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 \Delta_m 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 \Delta_3 is a triangle, whose vertices 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.


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