Existence of Balanced Simplices on Polytopes

The classic Sperner lemma states that in a simplicial subdivision of a simplex in Rn and a labelling rule satisfying some boundary condition there is a completely labeled simplex. In this paper we first generalize the concept of completely labeled simplex to the concept of a balanced simplex. Using this latter concept we then present a general combinatorial theorem, saying that under rather mild boundary conditions on a given labelling function there exists a balanced simplex for any given simplicial subdivision of a polytope. This theorem implies the well-known lemmas of Sperner, Scarf, Shapley, and Garcia as well as some other results as special cases. An even more general result is obtained when the boundary conditions on the labelling function are not required to hold. This latter result includes several results of Freund and Yamamoto as special cases.