Hardness of approximation of centered convex bodies by polytopes

TL;DR

This paper investigates the complexity of approximating centered convex bodies by polytopes, showing that non-symmetric bodies require significantly more vertices or facets for approximation within certain bounds.

Contribution

It establishes lower bounds on the number of vertices or facets needed to approximate general centered convex bodies, extending known results from symmetric cases.

Findings

Symmetric bodies can be approximated with fewer vertices within a factor of rac{n}{\u221a{\u03bb N}}.

General centered convex bodies require rac{n}{a0a0a0a0} more vertices or facets for similar approximation.

Lower bounds are valid for both vertex and facet approximations, with bounds depending on the dimension and number of vertices/facets.

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.