Suppose we have a convex polyhedron in $\Bbb R^n$ symmetric then we know there is an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it.
What is the minimal number of vertices of the polytope whose convex hull determines the ellipsoid exactly when the polyhedra's barycenter (arithmetic mean of vertices) agrees with center of ellipsoid?
Are there non-trivial cases where knowing $O(n^c)$ vertices suffices for some fixed $c>0$?