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 $S\subset {\mathbb{R}}^d$, then there are at most $2d$ points of $S$ whose convex hull contains the origin in the interior. Bárány, 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 $Q^{\mathrm{\prime }}$ satisfies $$r{\mathbf{B}}^d\subset Q^{\mathrm{\prime }}$$ with $r\ge d^{-2d}$. They conjectured that $r\ge c{d}^{-1\mathrm{/}2}$ holds with a universal constant $c>0$. We prove $r\ge \frac{1}{5d^2}$, the first polynomial lower bound on $r$. Furthermore, we show that $r$ is not be greater than $\frac{2}{\sqrt{d}}$.