Let K be a symmetric convex body in R N for which B 2 N is the ellipsoid of minimal volume. We provide estimates for the geometric distance of a ‘typical’ rank n projection of K to B 2 n , for 1⩽ n< N. Known examples show that the resulting estimates are optimal (up to numerical constants) even for the Banach–Mazur distance. To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 345–350.