The well known Sperner lemma states that in a simplicial subdivision of a simplex with a properly labeled boundary there is a completely labeled simplex. We present two combinatorial theorems on polytopes which generalize Sperner's lemma. Using balanced simplices, a generalized concept of completely labeled simplices, a uni ed existence result of balanced simplices in any simplicial subdivision of a polytope is given. This theorem implies the well-known lemmas of Sperner, Scarf, Shapley, and Garcia as well as some other results as special cases. A second theorem which imposes no restrictions on the integer labeling rule is established; this theorem implies several results of Freund.
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Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number
25.
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