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A complete description of convex bodies possessing the Markov property

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Abstract

A necessary and sufficient condition is derived (in terms of supporting functionals) for a closed convex body in a partially ordered Banach space to possess “Markov's property:” ifa < c < b,a ∈ K, b ∈ K, then c ∈ Int K. Our criterion enables us to express certain analytic results in geometrical form.

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Literature cited

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    M. G. Krein, “Some ideals of P. L. Chebyshev and A. A. Markov concerning limiting values of integrals and their further development,” Uspekhi Matem. Nauk,6, No. 4, 3–120 (1951).

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    A. A. Nudel'man, “A generalization of certain results obtained by P. L. Chebyshev and A. A. Markov,” Dokl. Akad. Nauk SSSR,125, No. 4, 740–742 (1959).

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Additional information

Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 733–740, December, 1970.

The author wishes to express his gratitude to M. G. Krein and the members of his seminar for their discussion of this work.

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Nudel'man, A.A. A complete description of convex bodies possessing the Markov property. Mathematical Notes of the Academy of Sciences of the USSR 8, 883–887 (1970). https://doi.org/10.1007/BF01673688

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Keywords

  • Banach Space
  • Convex Body
  • Markov Property
  • Geometrical Form
  • Order Banach Space