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.^[1]

The KKM lemma can be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem.

Statement

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.

Example

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.

References

↑ Knaster, B.; Kuratowski, C.; Mazurkiewicz, S. (1929), "Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe", Fundamenta Mathematicae (in German) 14 (1): 132–137 .

External links

See the proof of KKM Lemma in Planet Math.