A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it with barycenter of the polyhedron agreeing with center of the ellipsoid.

1. Does getting an uniform sample of $O(n^c)$ vertices for some fixed $c\geq2$ suffice to approximate the ellipsoid up to $\epsilon$ factor in each axis provided we know that the barycenter agrees with center of the ellipsoid and we know the center within some neighborhood?

I am asking this since the ellipsoid has only $O(n^2)$ parameters.