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Sperner’s Lemma and Brouwer’s Fixed-Point Theorem
| Content Provider | Semantic Scholar |
|---|---|
| Author | Shapiro, Joel H. |
| Copyright Year | 2015 |
| Abstract | These notes present a proof of the Brouwer Fixed-Point Theorem using a remarkable combinatorial lemma due to Emanuel Sperner. The method works in RN for all N, but for simplicity we’ll restrict the discussion to N = 2. Overview. In dimension two the Brouwer Fixed-Point Theorem states that every continuous mapping taking a closed disc into itself has a fixed point. Here we’ll give a proof of this special case of Brouwer’s result, but for triangles, rather than discs; closed triangles are homeomorphic to closed discs (Exercise 2.1 below) so our result will be equivalent to Brouwer’s. We’ll base our proof on an apparently unrelated combinatorial lemma due to Emanuel Sperner, which—in dimension two—concerns a certain method of labeling the vertices of “regular” decompositions of triangles into subtriangles. We’ll give two proofs of this special case of Sperner’s Lemma, one of which has come to serve as a basis for algorithms designed to approximate Brouwer fixed points. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://joelshapiro.org/Pubvit/Downloads/SpernerBrouwer/Sperner_Brouwer.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
