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

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGT_2006_26_3_a8,
     author = {Idzik, Adam and Junosza-Szaniawski, Konstanty},
     title = {Combinatorial lemmas for polyhedrons {I}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {439--338},
     publisher = {mathdoc},
     volume = {26},
     number = {3},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2006_26_3_a8/}
}
TY  - JOUR
AU  - Idzik, Adam
AU  - Junosza-Szaniawski, Konstanty
TI  - Combinatorial lemmas for polyhedrons I
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2006
SP  - 439
EP  - 338
VL  - 26
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2006_26_3_a8/
LA  - en
ID  - DMGT_2006_26_3_a8
ER  - 
%0 Journal Article
%A Idzik, Adam
%A Junosza-Szaniawski, Konstanty
%T Combinatorial lemmas for polyhedrons I
%J Discussiones Mathematicae. Graph Theory
%D 2006
%P 439-338
%V 26
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2006_26_3_a8/
%G en
%F DMGT_2006_26_3_a8
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/

[1] A.D. Alexandrov, Convex Polyhedra (Springer, Berlin, 2005).

[2] R.W. Freund, Variable dimension complexes Part II: A unified approach to some combinatorial lemmas in topology, Math. Oper. Res. 9 (1984) 498-509, doi: 10.1287/moor.9.4.498.

[3] C.B. Garcia, A hybrid algorithm for the computation of fixed points, Manag. Sci. 22 (1976) 606-613, doi: 10.1287/mnsc.22.5.606.

[4] B. Grunbaum, Convex Polytopes (Wiley, London, 1967).

[5] A. Idzik and K. Junosza-Szaniawski, Combinatorial lemmas for nonoriented pseudomanifolds, Top. Meth. in Nonlin. Anal. 22 (2003) 387-398.

[6] A. Idzik and K. Junosza-Szaniawski, Combinatorial lemmas for polyhedrons, Discuss. Math. Graph Theory 25 (2005) 95-102, doi: 10.7151/dmgt.1264.

[7] B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929) 132-137.

[8] W. Kulpa, Poincaré and Domain Invariance Theorem, Acta Univ. Carolinae - Mathematica et Physica 39 (1998) 127-136.

[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.

[10] H. Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math. 15 (1967) 1328-1343, doi: 10.1137/0115116.

[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.

[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.