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Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces

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Abstract

A closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here:

  1. (1)

    LetW be an arbitrary closed, convex and bounded body in a separable polyhedral Banach spaceE and let ε>0. Then there exists a tangential ε-approximating polytopeP for the bodyW.

  2. (2)

    LetP be a polytope in a separable Banach spaceE. Then, for every ε>0,P can be ε-approximated by an analytic, closed, convex and bounded bodyV.

We deduce from these two results that in a polyhedral Banach space (for instance in c0(ℕ) or inC(K) forK countable compact), every equivalent norm can be approximated by norms which are analytic onE/{0}.

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Authors and Affiliations

  1. Department of Mathematics, Université de Bordeaux, 351, cours de la libération, 33400, Talence, France

    Robert Deville

  2. Department of Mathematics and Computer Sciences, Ben Gurion University of the Negev, 84105, Beer Sheva, Israel

    Vladimir Fonf

  3. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Canada

    Petr Hájek

Authors
  1. Robert Deville
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  2. Vladimir Fonf
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  3. Petr Hájek
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Correspondence to Robert Deville.

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Deville, R., Fonf, V. & Hájek, P. Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces. Isr. J. Math. 105, 139–154 (1998). https://doi.org/10.1007/BF02780326

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  • DOI: https://doi.org/10.1007/BF02780326

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