### Abstract

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.

Original language | English |
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Place of Publication | Tilburg |

Publisher | Operations research |

Number of pages | 19 |

Volume | 1996-25 |

Publication status | Published - 1996 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 1996-25 |

### Keywords

- econometrics

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## Cite this

Eaves, C., van der Laan, G., Talman, A. J. J., & Yang, Z. F. (1996).

*Balanced Simplices on Polytopes*. (CentER Discussion Paper; Vol. 1996-25). Operations research.