Mass-to-charge ratio

The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz.

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 $\Delta _{m}$ 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

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External links

• See the proof of KKM Lemma in Planet Math.