Balanced Simplices on Polytopes

C. Eaves, G. van der Laan, A.J.J. Talman, Z.F. Yang

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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 languageEnglish
Place of PublicationTilburg
PublisherOperations research
Number of pages19
Publication statusPublished - 1996

Publication series

NameCentER Discussion Paper


  • econometrics

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    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.