Brouwer in Dimension Two
In dimension two the Brouwer Fixed-Point Theorem states that every continuous mapping taking a closed disc into itself has a fixed point. In this chapter 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 2.2 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.
KeywordsConvex Hull Triangle Inequality Interior Vertex Closed Disc Fair Division