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. Bárány, 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 \(\textbf{B}^d\) , there exist at most 2d vertices of Q whose convex hull \(Q'\) satisfies \(r\textbf{B}^d \subset Q' \) with \(r \ge d^{-2d}\) . Recently, Márton Naszódi 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 \(r \ge \frac{1}{5 \alpha d}\) . 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.
