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arXiv · 2512.22907

A point in the interior of the convex hulls

Imre Bárány·Yun Qi
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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.

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Imre Bárány, Yun Qi. 2026-06-18. A point in the interior of the convex hulls. https://arxiv.org/abs/2512.22907

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