Combinatorial lemmas for polyhedrons

Adam Idzik; Konstanty Junosza-Szaniawski

Discussiones Mathematicae Graph Theory (2005)

  • Volume: 25, Issue: 1-2, page 95-102
  • ISSN: 2083-5892

Abstract

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We formulate general boundary conditions for a labelling to assure the existence of a balanced n-simplex in a triangulated polyhedron. Furthermore we prove a Knaster-Kuratowski-Mazurkiewicz type theorem for polyhedrons and generalize some theorems of Ichiishi and Idzik. We also formulate a necessary condition for a continuous function defined on a polyhedron to be an onto function.

How to cite

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Adam Idzik, and Konstanty Junosza-Szaniawski. "Combinatorial lemmas for polyhedrons." Discussiones Mathematicae Graph Theory 25.1-2 (2005): 95-102. <http://eudml.org/doc/270665>.

@article{AdamIdzik2005,
abstract = {We formulate general boundary conditions for a labelling to assure the existence of a balanced n-simplex in a triangulated polyhedron. Furthermore we prove a Knaster-Kuratowski-Mazurkiewicz type theorem for polyhedrons and generalize some theorems of Ichiishi and Idzik. We also formulate a necessary condition for a continuous function defined on a polyhedron to be an onto function.},
author = {Adam Idzik, Konstanty Junosza-Szaniawski},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {KKM covering; labelling; primoid; pseudomanifold; simplicial complex; Sperner lemma},
language = {eng},
number = {1-2},
pages = {95-102},
title = {Combinatorial lemmas for polyhedrons},
url = {http://eudml.org/doc/270665},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Adam Idzik
AU - Konstanty Junosza-Szaniawski
TI - Combinatorial lemmas for polyhedrons
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 1-2
SP - 95
EP - 102
AB - We formulate general boundary conditions for a labelling to assure the existence of a balanced n-simplex in a triangulated polyhedron. Furthermore we prove a Knaster-Kuratowski-Mazurkiewicz type theorem for polyhedrons and generalize some theorems of Ichiishi and Idzik. We also formulate a necessary condition for a continuous function defined on a polyhedron to be an onto function.
LA - eng
KW - KKM covering; labelling; primoid; pseudomanifold; simplicial complex; Sperner lemma
UR - http://eudml.org/doc/270665
ER -

References

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  1. [1] T. Ichiishi and A. Idzik, Closed coverings of convex polyhedra, Internat. J. Game Theory 20 (1991) 161-169, doi: 10.1007/BF01240276. Zbl0761.52010
  2. [2] T. Ichiishi and A. Idzik, Equitable allocation of divisible goods, J. Math. Econom. 32 (1998) 389-400, doi: 10.1016/S0304-4068(98)00053-6. Zbl1016.91067
  3. [3] A. Idzik and K. Junosza-Szaniawski, Combinatorial lemmas for nonoriented pseudomanifolds, Top. Meth. in Nonlin. Anal. 22 (2003) 387-398. Zbl1038.05010
  4. [4] B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein beweis des fixpunktsatzes für n-dimensionale simplexe, Fund. Math. 14 (1929) 132-137. Zbl55.0972.01
  5. [5] W. Kulpa, Poincaré and domain invariance theorem, Acta Univ. Carolinae - Mathematica et Physica 39 (1998) 127-136. Zbl1007.54040
  6. [6] G. van der Laan, D. Talman and Z. Yang, Intersection theorems on polytypes, Math. Programming 84 (1999) 333-352. 
  7. [7] 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
  8. [8] 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
  9. [9] 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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