Let $A$ and $B$ be disjoint finite subsets of $\mathbb{R}^d$ for some $d$. Furthermore, assume that $|A\cup B|\geq d+3$, $conv(A)\cap conv(B)\neq\emptyset$, and points in $A\cup B$ are in general position, i.e., any $d+1$ points of them are affinely independent. Prove that there exist $A^{'}\subseteq A$ and $B^{'}\subseteq B$ such that $|A^{'}|+|B^{'}|= d+2$ and $conv(A^{'})\cap conv(B^{'})\neq\emptyset$.
Note: This is not a homework. I am just reading a paper. I did not understand one of the claims which I write here as a problem above.
I just proved that $\dim (conv(A)\cap conv(B))\geq 1$, maybe it could be useful!
This a direct consequence of Kirchberger's theorem which states: Let $A$ and $B$ be finite sets in $\mathbb{R}^n$ such that, for every subset $C$ of $n + 2$ or fewer points of $A\cup B$, the sets $A\cap C$ and $B\cap C$ can be strictly separated. Then $A$ and $B$ can be strictly separated.