Hardness of approximation of centered convex bodies by polytopes

Abstract: The distance between convex bodies (K, L \subseteq \R^n) is defined as [ d(K,L)= \inf \left{ λ\ge 1: \ L-x \subseteq T (K-y) \subseteq λ(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 $N \in (n, e<sup>{cn})$ vertices so that [ P \subseteq K \subseteq λP ] where (λ= 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.