Quantitative Steinitz theorem and polarity
Grigory Ivanov
TL;DR
The paper provides a quantitative bound in the Steinitz framework by relating the size of a point set to the radius $r$ of a ball contained in a subset of its convex hull. The authors introduce a polarity-based center-finding technique that, when combined with a dual (polar) perspective and an Atlantis-type transfer, yields a constructive way to prune to at most $2d$ vertices while preserving a guaranteed radius $r= rac{1}{2(m+d)+1}$ with ${f B}^d$ containment. A key technical ingredient is a weighted-maximum principle on polars that ensures the existence of an interior center $c$ for which the corresponding polar vertices sum to zero, enabling the bounded reduction. The results imply corollaries for linear-in-$d$ vertex counts and provide a somewhat weaker polynomial bound in the classical Steinitz setting, thereby linking duality, centrality, and combinatorial convex geometry in a unified framework.
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. 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 $\mathbf{B}^d$, 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, 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., $αd$, then one can select $2d$ vertices such that $r \geq \frac{1}{5 αd}$. 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.