Quantitative Steinitz theorem and polarity

The classical Steinitz theorem asserts that if the origin lies within the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points in $S$ whose convex hull contains the origin within its interior. B\'ar\'any, Katchalski, and Pach established a quantitative version of Steinitz's theorem, showing that for a convex polytope $Q$ in $\mathbb{R}^d$ containing the standard Euclidean unit ball $\mathbf{B}^d$, there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $r\mathbf{B}^d \subset Q'$ with $r \geq d^{-2d}$. Recently, M\'arton Nasz\'odi and the author derived a polynomial bound on $r$. This paper aims to establish a bound on $r$ based on the number of vertices of $Q.$ In other words, we demonstrate an effective method to remove several points from the original set $Q$ without significantly altering the bound on $r$. Specifically, if the number of vertices of $Q$ scales linearly with the dimension, i.e., $\alpha d$, then one can select $2d$ vertices such that $r \geq \frac{1}{5 \alpha d}$. The proof relies on a polarity trick, which may be of independent interest: we demonstrate the existence of a point $c$ in the interior of a convex polytope $P \subset \mathbb{R}^d$ such that the vertices of the polar polytope $(P-c)^\circ$ sum up to zero.