Balanced Simplices on Polytopes

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

Research output: Working paperDiscussion paperOther research output

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 Tilburg Operations research 19 1996-25 Published - 1996

Publication series

Name CentER Discussion Paper 1996-25

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Polytopes
Sperner's Lemma
Subdivision
Theorem
Imply
Polytope
Existence Results
Labeling
Lemma
Restriction
Generalise
Integer

• econometrics

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). Tilburg: Operations research.
Eaves, C. ; van der Laan, G. ; Talman, A.J.J. ; Yang, Z.F. / Balanced Simplices on Polytopes. Tilburg : Operations research, 1996. (CentER Discussion Paper).
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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.",
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Eaves, C, van der Laan, G, Talman, AJJ & Yang, ZF 1996 'Balanced Simplices on Polytopes' CentER Discussion Paper, vol. 1996-25, Operations research, Tilburg.

Balanced Simplices on Polytopes. / Eaves, C.; van der Laan, G.; Talman, A.J.J.; Yang, Z.F.

Tilburg : Operations research, 1996. (CentER Discussion Paper; Vol. 1996-25).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Balanced Simplices on Polytopes

AU - Eaves, C.

AU - van der Laan, G.

AU - Talman, A.J.J.

AU - Yang, Z.F.

N1 - Pagination: 19

PY - 1996

Y1 - 1996

N2 - 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.

AB - 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.

KW - econometrics

M3 - Discussion paper

VL - 1996-25

T3 - CentER Discussion Paper

BT - Balanced Simplices on Polytopes

PB - Operations research

CY - Tilburg

ER -

Eaves C, van der Laan G, Talman AJJ, Yang ZF. Balanced Simplices on Polytopes. Tilburg: Operations research. 1996. (CentER Discussion Paper).