Best uniform approximation of a convex compact set by a ball in an arbitrary norm

The finite-dimensional problem of the best uniform approximation of a convex compact set by a ball with respect to an arbitrary norm in the Hausdorff metric corresponding to that norm is considered. When the compact set to be estimated and the norm ball are polyhedra, the problem is shown to reduce to a linear program. This fact is used to design an iterative method for solving the problem in the case of an arbitrary compact set and an arbitrary norm. At every step of the method, the unit ball in the norm used and the underlying compact set are replaced by their outer polyhedral approximations, where the polyhedra are constructed from supporting hyperplanes drawn through certain boundary points.