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On extreme points of convex compact turing located set

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Logical Foundations of Computer Science (LFCS 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 813))

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Abstract

We are interested in determining the effective content of the Krein-Milman theorem. In this paper we discuss the finite dimensional case. We design an algorithm,(Algorithm E) to locate the set of extreme points of a convex compact Turing located set K ⫅R q. A sequence of approximating convex hulls for K is computed. Several lemmas about the extreme points of a convex compact set K are proved to guarantee the correctness of the algorithm. The condition that K is Turing located is necessary for the success of Algorithm E, that is, Algorithm E is effective if and only if K is Turing located. Algorithm E ia an effective version of the finite dimensinal Krein-Milman Theorem. A sequel will investigate the effective content of the KreinMilman theorem in effectively presented metric spaces.

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References

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Authors and Affiliations

  1. Mathematical Sciences Institute, Cornell University-Ithaca, 14850, NY, USA

    Xiaolin Ge & Anil Nerode

Authors
  1. Xiaolin Ge
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  2. Anil Nerode
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Anil Nerode Yu. V. Matiyasevich

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© 1994 Springer-Verlag Berlin Heidelberg

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Ge, X., Nerode, A. (1994). On extreme points of convex compact turing located set. In: Nerode, A., Matiyasevich, Y.V. (eds) Logical Foundations of Computer Science. LFCS 1994. Lecture Notes in Computer Science, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58140-5_12

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  • DOI: https://doi.org/10.1007/3-540-58140-5_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58140-6

  • Online ISBN: 978-3-540-48442-4

  • eBook Packages: Springer Book Archive

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