John’s position is not good for approximation

Abstract

A well-known consequence of John’s theorem states that any symmetric convex body K ⊂ ℝⁿ in John’s position can be approximated by a polytope P with a polynomial number of facets in n, so that $P\; \subset \;K\; \subset \;\sqrt n P$. This results extends to the non-symmetric case if the homothety ratio grows to n. In this note, we study how well this result holds in the non-symmetric case, if the homothety ratio is reduced below n. We prove the following: For R = o(n) and a sufficiently large n, there exists a convex body K ⊂ ℝⁿ in John’s position for which there is no polytope P with a polynomial in n number of facets, such that K ⊂ P ⊂ RK. Moreover, for $R = O\left({\sqrt n} \right)$, there exists a convex body for which a polytope with an exponential number of facets is needed.