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Topological degree and the Sperner lemma

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In this paper, we give a new proof of Sperner's lemma, in its superstrong from, using the topological degree. Thus, we point out a relation between several methods for fixed-point theorems using either the topological degree, or the KKM lemma, or the Sperner lemma.

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  1. 1.

    Schwartz, J. T.,Nonlinear Functional Analysis, Gordon and Breach, New York, New York, 1969.

  2. 2.

    Lasry, J. M., andRobert, R.,Degré et Théorèmes de Point Fixe pour les Applications Multivoques et Applications, Cahier de Mathématique de la Décision, Université Paris-IX, Dauphine, France, 1975.

  3. 3.

    Geistdoerfer-Florenzano, M.,L'Equilibre Economique Général Transitif et Intransitif—Problèmes d'Existence, Centre d'Etudes Prospectives d'Economie Mathématique Appliquées à la Planification, Paris, France, Report No. 8004, 1980.

  4. 4.

    Scarf, H.,The Approximation of Fixed Points of a Continuous Mapping, SIAM Journal on Applied Mathematics, Vol. 15, No. 5, 1967.

  5. 5.

    Kuhn, H. W., andmacKinnon, J. G.,Sandwich Methods for Finding Fixed Points, Journal of Optimization Theory and Applications, Vol. 17, Nos. 3/4, 1975.

  6. 6.

    Hoang Tuy,Pivotal Methods for Computing Equilibrium Points: Unified Approach and New Restart Algorithm, Mathematical Programming, Vol. 16, pp. 210–227, 1979.

  7. 7.

    Todd, M. J.,The Computation of Fixed Points and Applications, Springer-Verlag, Berlin, Germany, 1976.

  8. 8.

    Berge, C.,Espaces Topologiques—Fonctions Multivoques, Dunod, Paris, France, 1959.

  9. 9.

    Yoseloff, M.,Topologic Proofs of Some Combinatorial Theorems, Journal of Combinatorial Theory, (A.), Vol. 17, pp. 95–111, 1974.

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The author would like to thank Dr. G. Leitmann for his remarks and suggestions.

Communicated by G. Leitmann

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Le Van, C. Topological degree and the Sperner lemma. J Optim Theory Appl 37, 371–377 (1982).

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Key Words

  • Topological degree
  • simplex
  • triangulations