Statement of the lemma
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.
- Knaster, B.; Kuratowski, C.; Mazurkiewicz, S. (1929), "Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe", Fundamenta Mathematicae (in German) 14 (1): 132–137.
- See the proof of KKM Lemma in Planet Math.