Hardness of approximation of centered convex bodies by polytopes

The distance between convex bodies $K, L \subseteq \R^n$ is defined as $$d(K,L)= \inf \left\{ \lambda \ge 1: \ L-x \subseteq T (K-y) \subseteq \lambda (L-x) \right\},$$ where the infimum is taken over all $x,y \in \R^n$ and all invertible linear operators $T: \R^n \to \R^n$. If both bodies are centrally symmetric, then the shifts $x$ and $y$ can be chosen to be $0$. In this case, any convex symmetric body $K$ can be approximated by a polytope $P$ with at most $N \in (n, e^{cn})$ vertices so that $$P \subseteq K \subseteq \lambda P$$ where $\lambda= O \left(\sqrt{\frac{n}{\log N}} \right)$ up to logarithmic factors. We prove that approximating a general centered convex body by a polytope requires a significantly larger number of vertices compared to the symmetric case. More precisely, there exists a convex body $K \subseteq \R^n$ whose barycenter coincides with the origin, such that any polytope $P$ satisfying $$P \subseteq K \subseteq c \, \frac{n}{\log N} P$$ must have at least $N$ vertices, provided that $N \in (Cn^2, e^{cn})$. Moreover, we prove that the same bound holds for approximating a centered convex body with a polytope having $N$ facets instead of $N$ vertices.