A point in the interior of the convex hulls

Abstract

Steinitz's theorem states that if a point $a\in \mathrm{int}\mathrm{conv}X$ for a set $X\subset {\mathbb{R}}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a\in \mathrm{int}\mathrm{conv}Y$. 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.