# John's Position is not good for approximation

## TL;DR

The paper demonstrates limitations in approximating convex bodies in John's position with polytopes, showing that certain bodies cannot be closely approximated with polytopes having polynomial or subexponential facets.

## Contribution

It establishes new lower bounds on the complexity of polytope approximations for convex bodies in John's position, highlighting fundamental limitations.

## Key findings

- No polynomial-facet polytope can approximate certain convex bodies within a factor R_n
- No subexponential-facet polytope can approximate some convex bodies within a factor of n
- Results apply to high-dimensional convex geometry and approximation theory

## 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 $\lim_{n\rightarrow \infty} \frac{R_n}{n}=0$. 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 $K\subset P\subset R_nK$; 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 $K\subset P \subset \sqrt{n}K$.