Contact points of convex bodies

LetB be a convex body in ℝn and let ɛ be an ellipsoid of minimal volume containingB. By contact points ofB we mean the points of the intersection between the boundaries ofB and ɛ. By a result of P. Gruber, a generic convex body in ℝn has (n+3)·n/2 contact points. We prove that for every ɛ>0 and for every convex bodyB ⊂ ℝn there exists a convex bodyK having\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$m \leqslant C(\varepsilon ) \cdot n\log ^3 n$$ \end{document} contact points whose Banach-Mazur distance toB is less than 1+ɛ.