  Recall that a convex body $K$ is in John's position if the unit Euclidean
ball is the maximal volume ellipsoid contained in $K$. Approximating convex
body in John's position by polytopes we obtain the following results. 1. Let
$n>R_n\ge 1$ be a sequence such that $\lim_{n\rightarrow \infty}
\frac{R_n}{n}=0$. For a sufficiently large $n$, we can construct a convex body
$K\subset \mathbb{R}^n$ in John's position such that there is no $P$, polytope
with a polynomial number of facets in $n$ such that $K\subset P\subset R_nK$;
2. For a sufficiently large $n$, there is a convex body $K\subset \mathbb{R}^n$
in John's position such that there is no $P$, polytope that has less than
$\exp(cn)$ facets satisfies $K\subset P \subset \sqrt{n}K$.
