I am writing about (convex) polyhedra (in 3-dimensional Euclidean space) and am trying to determine some symmetry groups. It would be very good to be able to say that every polyhedron has a unique sphere with minimal radius containing it. If this were true, then the center of this sphere would be preserved by the symmetries of the polyhedron (because they preserve distances) and these symmetries would be easier to determine (since they would have a fixed point).
Now, I have two related questions and I would appreciate also partial answers:
(1) Does such a unique minimal sphere always exist? Is there an appropriate restriction on the type of polyhedra considered that guarantees its existence?
(2) If such a sphere exists, it cannot, in general, touch all vertices of the polyhedron. So can the sphere be characterized in other ways? What kind of point is the center of this sphere?