Quantitative Steinitz Theorem: A polynomial bound

Abstract: 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 $\frac{2}{\sqrt{d}}$.