Balanced Simplices on Polytopes

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.