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

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) LetW be an arbitrary closed, convex and bounded body in a separable polyhedral Banach spaceE and let e>0. Then there exists a tangential e-approximating polytopeP for the bodyW. (2) LetP be a polytope in a separable Banach spaceE. Then, for every e>0,P can be e-approximated by an analytic, closed, convex and bounded bodyV.