Geodesic
Combinatorial lemmas for polyhedrons I
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 3, pp. 439-338

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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.
Keywords: b-balanced simplex, labelling, polyhedron, simplicial complex, Sperner lemma 
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     title = {Combinatorial lemmas for polyhedrons {I}},
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Idzik, Adam; Junosza-Szaniawski, Konstanty. Combinatorial lemmas for polyhedrons I. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 3, pp. 439-338. http://geodesic.mathdoc.fr/item/DMGT_2006_26_3_a8/