Explicit polyhedral approximation of the Euclidean ball

We discuss the problem of computing points of I Rⁿ whose convex hull contains the Euclidean ball, and is contained in a small multiple of it. Given a polytope containing the Euclidean ball, we introduce its successor obtained by intersection with all tangent spaces to the Euclidean ball, whose normals point towards the vertices of the polytope. Starting from the L^∞ ball, we discuss the computation of the two first successors, and give a complete analysis in the case when n=6.