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A constructive proof of a permutation-based generalization of Sperner’s lemma. (English) Zbl 0673.55004
Summary: In a recent paper, D. Gale [Int. J. Game Theory 13, 61-64 (1984; Zbl 0531.90011)] has given an interesting generalization of the KKM lemma in combinatorial topology. We present a similar generalization of Sperner’s well-known lemma and give a constructive proof. The argument uses the familiar idea of following simplicial paths in a triangulation. To demonstrate that the algorithm must work, orientation considerations are necessary. Gale’s generalized KKM lemma is derived from the main result. A permutation-based generalization of Brouwer’s fixed point theorem is also given.

MSC:
55M99 Classical topics in algebraic topology
57Q99 PL-topology
55M20 Fixed points and coincidences in algebraic topology
55M10 Dimension theory in algebraic topology
05B99 Designs and configurations
05C99 Graph theory
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References:
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