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Intersection theorems with a continuum of intersection points


Author Info

  • Herings, P.J.J.
  • Talman, A.J.J.

    (Tilburg University, Center for Economic Research)


In all existing intersection theorems conditions are given under which acertain subset of acollection of sets has a non-empty intersection. In this paper conditions are formulated under which the intersection is a continuum of points satisfying some interesting topological properties. In this sense the intersection theorems considered in this paper belong to a new class. The intersection theorems are formulated on the unit cube and it is shown that both the vector of zeroes and the vector of ones lie in the same component of the intersection. This is interesting for some specific applications. The theorems give a generalization of the well-known lemmas of Knaster, Kuratowski, and Mazurkiewicz, of Sperner, of Shapley, and of Ichiischi. Moreover the results can be used to sharpen the usual formulation of the Sperner Lemma on the cube.

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Bibliographic Info

Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 1994-79.

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Date of creation: 1994
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Handle: RePEc:dgr:kubcen:199479

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Keywords: Optimization; operations research;


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  1. Shapley, Lloyd & Vohra, Rajiv, 1991. "On Kakutani's Fixed Point Theorem, the K-K-M-S Theorem and the Core of a Balanced Game," Economic Theory, Springer, vol. 1(1), pages 108-16, January.
  2. Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 1999. "Intersection theorems on polytypes," Open Access publications from Tilburg University urn:nbn:nl:ui:12-78480, Tilburg University.
  3. Herings, P.J.J., 1992. "On the structure of constrained equilibria," Research Memorandum 587, Tilburg University, Faculty of Economics and Business Administration.
  4. Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer, vol. 20(2), pages 161-69.
  5. van der Laan, G. & Talman, D., 1993. "Intersection Theorems on the Simplotope," Papers 9370, Tilburg - Center for Economic Research.
  6. Dreze, Jacques H, 1975. "Existence of an Exchange Equilibrium under Price Rigidities," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 16(2), pages 301-20, June.
  7. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
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Cited by:
  1. Talman, A.J.J. & Yamamoto, M., 2004. "Contimuum of zero points of a mapping on a compact, convex set," Open Access publications from Tilburg University urn:nbn:nl:ui:12-141075, Tilburg University.
  2. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions: Existence and Computation," Discussion Paper 1999-72, Tilburg University, Center for Economic Research.
  3. Herings, P.J.J. & Koshevoy, G.A. & Talman, A.J.J. & Yang, Z.F., 2002. "A General Existence Thorem of Zero Points," Discussion Paper 2002-107, Tilburg University, Center for Economic Research.


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