  The classical Steinitz theorem states that if the origin belongs to the
interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are
at most $2d$ points of $S$ whose convex hull contains the origin in the
interior. B\'ar\'any, Katchalski, and Pach proved the following quantitative
version of Steinitz's theorem. Let $Q$ be a convex polytope in $\mathbb{R}^d$
containing the standard Euclidean unit ball $\mathbf{B}^d$. Then there exist at
most $2d$ vertices of $Q$ whose convex hull $Q^\prime$ satisfies \[ r
\mathbf{B}^d \subset Q^\prime \] with $r\geq d^{-2d}$. They conjectured that
$r\geq c d^{-1/2}$ holds with a universal constant $c>0$. We prove $r \geq
\frac{1}{5d^2}$, the first polynomial lower bound on $r$. Furthermore, we show
that $r$ is not be greater than $\frac{2}{\sqrt{d}}$.
