TY - JOUR

T1 - Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces

AU - Deville, Robert

AU - Fonf, Vladimir

AU - Hájek, Petr

PY - 1998

Y1 - 1998

N2 - A closed, convex and bounded set P in a Banach space E is called a polytope if every finite-dimensional section of P is a polytope. A Banach space E is called polyhedral if E has an equivalent norm such that its unit ball is a polytope. We prove here : (1) Let W be an arbitrary closed, convex and bounded body in a separable polyhedral Banach space E and let ε > 0. Then there exists a tangential ε-approximating polytope P for the body W. (2) Let P be a polytope in a separable Banach space E. Then, for every ε > 0, P can be ε-approximated by an analytic, closed, convex and bounded body V. We deduce from these two results that in a polyhedral Banach space (for instance in c
0(ℕ) or in C(K) for K countable compact), every equivalent norm can be approximated by norms which are analytic on E\{0}.

AB - A closed, convex and bounded set P in a Banach space E is called a polytope if every finite-dimensional section of P is a polytope. A Banach space E is called polyhedral if E has an equivalent norm such that its unit ball is a polytope. We prove here : (1) Let W be an arbitrary closed, convex and bounded body in a separable polyhedral Banach space E and let ε > 0. Then there exists a tangential ε-approximating polytope P for the body W. (2) Let P be a polytope in a separable Banach space E. Then, for every ε > 0, P can be ε-approximated by an analytic, closed, convex and bounded body V. We deduce from these two results that in a polyhedral Banach space (for instance in c
0(ℕ) or in C(K) for K countable compact), every equivalent norm can be approximated by norms which are analytic on E\{0}.

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U2 - 10.1007/BF02780326

DO - 10.1007/BF02780326

M3 - Article

SN - 0021-2172

VL - 105

SP - 139

EP - 154

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -