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The extreme point characterization of semi-infinite dual non-Archimedean balls

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Abstract

The extreme point characterization of the (l′)-ball of a generalized finite sequence space by Kortanek and Strojwas was accomplished only for real scalars and by continuity considerations. We show that no topology or continuity is needed as in Kortanek-Strojwas and that the characterization extends to weighted (l′)-balls with any ordered scalar field. We show a Chebyshev ball theorem is false since they haveno extreme points. Via generalizing the LIEP theorem, useful projections of the ball are proved convex hulls of their extreme points.

Zusammenfassung

Die Extrempunkt-Charakterisierung der (l′)-Kugel eines verallgemeinerten endlichen Folgenraumes ist von Kortanek und Strojwas nur für reelle Skalare und mittels Stetigkeitsbetrachtungen hergeleitet worden. In der vorliegenden Arbeit wird gezeigt, daß keine topologischen oder Stetigkeitsbetrachtungen wie bei Kortanek und Strojwas erforderlich sind und daß die Kortanek-Strojwas-Charakterisierung auch für gewichtete (l′)-Kugeln und einen beliebig angeordneten Körper gültig bleibt. Ein analoges Theorem für Tschebyschew-Kugeln ist falsch, da diese Kugeln keine Extrempunkte besitzen. Durch Verallgemeinerung des LIEP-Theorems (lineare Unabhängigkeit bei Extrempunkten) der semi-infinite Optimierung erhält man Charakterisierungen von gewissen Untermengen der Tschebyschew-Kugel als konvexe Hüllen ihrer Extrempunkte.

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References

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Authors and Affiliations

  1. Center for Cybernetic Studies, Graduate School of Business, 4.138, The University of Texas at Austin, 78712, Austin, Texas

    A. Charnes &  T. Song

Authors
  1. A. Charnes
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  2. T. Song
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Charnes, A., Song, T. The extreme point characterization of semi-infinite dual non-Archimedean balls. Zeitschrift für Operations Research 28, 101–115 (1984). https://doi.org/10.1007/BF01918768

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  • DOI: https://doi.org/10.1007/BF01918768

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