Let $A_1,\dots,A_{d+1}$ be finite subsets of $\mathbb R^d$. If $0\in \operatorname{conv}(A_i)$ for every $i$, then there exist points $a_i\in A_i$ such that
\begin{align*}
0\in \operatorname{conv}\{a_1,\dots,a_{d+1}\}.
\end{align*}
Knowledge Status
Discrete MathematicsCombinatorics
Discussion
Colorful Caratheodory Theorem: Let A_1,,A_d+1 be finite subsets of R^d.