Knaster–Kuratowski–Mazurkiewicz lemma
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
The KKM lemma says that a KKM covering has a non-empty intersection, i.e:
- .
Example
covers vertex 1, covers vertex 2, covers vertex 3.
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 union of all three sets covers the entire triangle
The lemma is illustrated by the picture on the right, in which set #1 is blue, set #2 is red and set #3 is green. The KKM requirements are satisfied, since:
Each vertex is covered by a unique color.
Each edge is covered by the two colors of its two vertices.
The triangle is covered by all three colors.
The KKM lemma states that there is a point covered by all three colors simultaneously; such a point is clearly visible in the picture.
Equivalent results
There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column.[2]
Algebraic topology | Combinatorics | Set covering |
---|---|---|
Brouwer fixed-point theorem | Sperner's lemma | Knaster–Kuratowski–Mazurkiewicz lemma |
Borsuk–Ulam theorem | Tucker's lemma | Lusternik–Schnirelmann theorem |
Generalizations
Permutations
- .
Gale wrote about his lemma: "A colloquial statement of this result is the red, white and blue lemma which asserts that if each of three people paint a triangle red, white and blue according to the KKM rules, then there will be a point which is in the red set of one person, the white set of another, the blue of the third".[3]
Ravindra Bapat provided an alternative proof of this generalization, based on the permutation variant of Sperner's lemma.[4]
Note: the original KKM lemma follows from the permutation lemma by simply picking n identical coverings.
Connecting-sets
A connecting set of a simplex is a connected set that touches all n faces of the simplex.
The connecting-set variant also has a permutation variant, so that both these generalizations can be used simultaneously.
The KKMS theorem
The KKMS theorem is a generalization of the KKM lemma by Lloyd Shapley. It is useful in economics, especially in cooperative game theory.[6]
The collection {{1}, {2}, {3}} is balanced: choose all weights to be 1. The same is true for any collection in which each element appears exactly once, such as the collection {{1,2},{3}} or the collection { {1,2,3} }.
The collection {{1,2}, {2,3}, {3,1}} is balanced: choose all weights to be 1/2. The same is true for any collection in which each element appears exactly twice.
The collection {{1,2}, {2,3}} is not balanced, since for any choice of positive weights, the sum for element 2 will be larger than the sum for element 1 or 3, so it is not possible that all sums equal 1.
The collection {{1,2}, {2,3}, {1}} is balanced: choose .
whenever ( is a singleton that contains only element ).
otherwise.
Reny and Wooders proved that the balanced set can also be chosen to be partnered.[11]