Quantitative Steinitz theorem: A polynomial bound

The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set S subset of R-d, then there are at most 2d points of S whose convex hull contains the origin in the interior. Barany, Katchalski, and Pach proved the following quantitative version of Steinitz's theorem. Let Q be a convex polytope in R-d containing the standard Euclidean unit ball B-d. Then there exist at most 2d vertices of Q whose convex hull Q ' satisfies rB(d) subset of Q ' with r >= d(-2d). They conjectured that r >= cd(-1/2) holds with a universal constant c > 0. We prove r >= 1/5d(2), the first polynomial lower bound on r. Furthermore, we show that r is not greater than 2/root d.