Hardness of approximation of centered convex bodies by polytopes

Abstract

The distance between convex bodies \(K, L \subseteq \R^n\) is defined as \[ d(K,L)= \inf \left\{ \lambda \ge 1: \ L-x \subseteq T (K-y) \subseteq \lambda (L-x) \right\}, \] 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.