Intersection of two convex sets

Question

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!

Please add your thoughts on this question. Also it would be nice to see some words regarding Radon's theorem in this post, since it is pretty similar to your question. — mike239x, Mar 10 '18 at 19:51
@szw17101: I think with Carathedeory's theorem just we can assume that the size of $A$ and $B$ are less than or equal $d+1$. — MathFun, Mar 10 '18 at 20:14
Or... an element of $conv(A)\cap conv(B)$ is a convex combination of at most $d+1$ points from $A\cup B$. Take $A_1$ the set of these points from $A$ and $B_1$ similarly. If convex hulls of them do intersect, we are done. If not... we have at least two additional points.
This is my attempt only. I don't know whether it is good... — szw1710, Mar 10 '18 at 20:18

score 2 · Accepted Answer · answered Mar 12 '18 at 10:08

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.

Intersection of two convex sets

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