Title: Quantitative Steinitz theorem and polarity

Abstract: 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 the original polytope. In other words, we demonstrate an effective method to remove several points from the original set without significantly altering the bound on $r$. Specifically, if the number of vertices of $Q$ scales linearly with the dimension, i.e., $cd$, then one can select $2d$ vertices such that $r \geq \frac{1}{5cd}$. The proof relies on a polarity trick, which may be of independent interest: we demonstrate the existence of a point $p$ 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.