A point in the interior of the convex hulls

TL;DR

This paper extends Steinitz's theorem by proving a colorful version that guarantees a subset with specific properties, and characterizes cases requiring the maximum number of sets, advancing understanding of convex hulls in high dimensions.

Contribution

The authors establish a colorful variant of Steinitz's theorem and identify conditions when the maximum subset size of 2d is necessary.

Findings

Proved a colorful version of Steinitz's theorem.

Characterized cases requiring 2d sets.

Enhanced understanding of convex hulls in Euclidean space.

Abstract

Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here. We prove the colourful version of this theorem and characterize the cases when exactly $2d$ sets are needed.