John's Position is not good for approximation

Abstract

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 ${\lim }_{n\to \infty }\frac{R_n}{n}=0$. 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 $K\subset P\subset R_{n}K$; 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 $K\subset P\subset \sqrt{n}K$.