# Combinatorial lemmas for polyhedrons I

Adam Idzik; Konstanty Junosza-Szaniawski

Discussiones Mathematicae Graph Theory (2006)

- Volume: 26, Issue: 3, page 439-338
- ISSN: 2083-5892

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topAdam Idzik, and Konstanty Junosza-Szaniawski. "Combinatorial lemmas for polyhedrons I." Discussiones Mathematicae Graph Theory 26.3 (2006): 439-338. <http://eudml.org/doc/270349>.

@article{AdamIdzik2006,

abstract = {We formulate general boundary conditions for a labelling of vertices of a triangulation of a polyhedron by vectors to assure the existence of a balanced simplex. The condition is not for each vertex separately, but for a set of vertices of each boundary simplex. This allows us to formulate a theorem, which is more general than the Sperner lemma and theorems of Shapley; Idzik and Junosza-Szaniawski; van der Laan, Talman and Yang. A generalization of the Poincaré-Miranda theorem is also derived.},

author = {Adam Idzik, Konstanty Junosza-Szaniawski},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {b-balanced simplex; labelling; polyhedron; simplicial complex; Sperner lemma; convex polytope; triangulation; Sperner's lemma; balanced simplex},

language = {eng},

number = {3},

pages = {439-338},

title = {Combinatorial lemmas for polyhedrons I},

url = {http://eudml.org/doc/270349},

volume = {26},

year = {2006},

}

TY - JOUR

AU - Adam Idzik

AU - Konstanty Junosza-Szaniawski

TI - Combinatorial lemmas for polyhedrons I

JO - Discussiones Mathematicae Graph Theory

PY - 2006

VL - 26

IS - 3

SP - 439

EP - 338

AB - We formulate general boundary conditions for a labelling of vertices of a triangulation of a polyhedron by vectors to assure the existence of a balanced simplex. The condition is not for each vertex separately, but for a set of vertices of each boundary simplex. This allows us to formulate a theorem, which is more general than the Sperner lemma and theorems of Shapley; Idzik and Junosza-Szaniawski; van der Laan, Talman and Yang. A generalization of the Poincaré-Miranda theorem is also derived.

LA - eng

KW - b-balanced simplex; labelling; polyhedron; simplicial complex; Sperner lemma; convex polytope; triangulation; Sperner's lemma; balanced simplex

UR - http://eudml.org/doc/270349

ER -

## References

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- [9] G. van der Laan, D. Talman and Z. Yang, Existence of balanced simplices on polytopes, J. Combin. Theory (A) 96 (2001) 25-38, doi: 10.1006/jcta.2001.3178. Zbl1091.52007
- [10] H. Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math. 15 (1967) 1328-1343, doi: 10.1137/0115116. Zbl0153.49401
- [11] L.S. Shapley, On balanced games without side payments, in: T.C. Hu and S.M. Robinson (eds.), Mathematical Programming, New York: Academic Press (1973) 261-290. Zbl0267.90100
- [12] E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebiets, Abh. Math. Sem. Univ. Hamburg 6 (1928) 265-272, doi: 10.1007/BF02940617. Zbl54.0614.01

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