Quantitative Steinitz Theorem: A polynomial bound

TL;DR

This paper establishes a polynomial lower bound for a quantitative version of Steinitz's theorem, improving understanding of convex polytopes containing the Euclidean ball and their vertex configurations.

Contribution

The authors prove the first polynomial lower bound on the radius in a quantitative Steinitz theorem, advancing the theoretical understanding of convex polytopes in high dimensions.

Findings

Proved that r ≥ 1/(5d^2), providing a polynomial lower bound.

Showed that r cannot be greater than 2/√d, establishing an upper bound.

Enhanced the understanding of convex hulls containing the Euclidean ball in high dimensions.

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