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:
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(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.
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(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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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