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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. The KKM lemma can be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem. Statement Let \Delta_ be an (n-1)-dimensional simplex with ''n'' vertices labeled as 1,\ldots,n. A KKM covering is defined as a set C_1,\ldots,C_n of closed sets such that for any I \subseteq \, the convex hull of the vertices corresponding to I is covered by \bigcup_C_i. The KKM lemma says that in every KKM covering, the common intersection of all ''n'' sets is nonempty, i.e: :\bigcap_^n C_i \neq \emptyset. Example When n=3, the KKM lemma considers the simplex \Delta_2 which is a triangle, whose vertices can be labeled 1, 2 and 3. We are given three closed sets C_1,C_2,C_3 such that: * C_1 covers vertex 1, C_2 covers vertex 2, C_3 covers vertex 3. * The edge 12 (from vertex 1 to vertex 2) is covered by the se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Theorem
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors claim that results of this kind are amongst the most generally useful in mathematics. In mathematical analysis The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed-point theorem (1911) is a non- constructive result: it says that any continuous function from the closed unit ball in ''n''-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma). For example, the cosine function is continuous in ˆ’1,1and maps it into ˆ’1, 1 and thus must have a fixed point. This is clear when examining a sketched graph of the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
