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

Abstract

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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.

How to cite

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Adam 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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  1. [1] A.D. Alexandrov, Convex Polyhedra (Springer, Berlin, 2005). 
  2. [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. Zbl0556.57015
  3. [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. Zbl0358.90032
  4. [4] B. Grunbaum, Convex Polytopes (Wiley, London, 1967). 
  5. [5] A. Idzik and K. Junosza-Szaniawski, Combinatorial lemmas for nonoriented pseudomanifolds, Top. Meth. in Nonlin. Anal. 22 (2003) 387-398. Zbl1038.05010
  6. [6] A. Idzik and K. Junosza-Szaniawski, Combinatorial lemmas for polyhedrons, Discuss. Math. Graph Theory 25 (2005) 95-102, doi: 10.7151/dmgt.1264. Zbl1075.52502
  7. [7] B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929) 132-137. Zbl55.0972.01
  8. [8] W. Kulpa, Poincaré and Domain Invariance Theorem, Acta Univ. Carolinae - Mathematica et Physica 39 (1998) 127-136. Zbl1007.54040
  9. [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. [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. [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. [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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