A constructive proof of a permutation-based generalization of Sperner’s lemma

In a recent paper, Gale has given an interesting generalization of the KKM lemma in combinatorial topology. This paper presents a similar generalization of Sperner’s well-known lemma and gives 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 results. A permutation-based generalization of Brouwer’s fixed point theorem is also given.