Israel Journal of Mathematics

, Volume 105, Issue 1, pp 139–154 | Cite as

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



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}.


Banach Space Unit Ball Convex Body Implicit Function Theorem Equivalent Norm 
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Copyright information

© Hebrew University 1998

Authors and Affiliations

  1. 1.Department of MathematicsUniversité de BordeauxTalenceFrance
  2. 2.Department of Mathematics and Computer SciencesBen Gurion University of the NegevBeer ShevaIsrael
  3. 3.Department of MathematicsUniversity of AlbertaEdmontonCanada

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