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Maximal Points of Convex Sets in Locally Convex Topological Vector Spaces: Generalization of the Arrow–Barankin–Blackwell Theorem

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  • L.W. Woo

    (HCM Development)

  • R.K. Goodrich

    (University of Colorado)

Abstract

In 1953, Arrow, Barankin, and Blackwell proved that, if C is a nonempty compact convex set in Rn with its standard ordering, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. In this paper, we present a generalization of this result. We show that that, if C is a compact convex set in a locally convex topological space X and if K is an ordering cone on X such that the quasi-interiors of K and the dual cone K* are nonempty, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. For example, our work shows that, under the appropriate conditions, the density results hold in the spaces Rn, Lp(Ω, μ), 1≤p≤∞, lp, 1≤p≤∞, and C (Ω), Ω a compact Hausdorff space, when they are partially ordered with their natural ordering cones.

Suggested Citation

  • L.W. Woo & R.K. Goodrich, 2003. "Maximal Points of Convex Sets in Locally Convex Topological Vector Spaces: Generalization of the Arrow–Barankin–Blackwell Theorem," Journal of Optimization Theory and Applications, Springer, vol. 116(3), pages 647-658, March.
    • Handle: RePEc:spr:joptap:v:116:y:2003:i:3:d:10.1023_a:1023021604751
    DOI: 10.1023/A:1023021604751
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    References listed on IDEAS

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    1. Majumdar, Mukul, 1972. "Some general theorems on efficiency prices with an infinite-dimensional commodity space," Journal of Economic Theory, Elsevier, vol. 5(1), pages 1-13, August.
    2. Majumdar, Mukul, 1970. "Some approximation theorems on efficiency prices for infinite programs," Journal of Economic Theory, Elsevier, vol. 2(4), pages 399-410, December.
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    Cited by:

    1. Joseph Newhall & Robert K. Goodrich, 2015. "On the Density of Henig Efficient Points in Locally Convex Topological Vector Spaces," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 753-762, June.

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