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

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}.