Non-empty intersection of two convex hulls

Question

Suppose $A$ and $B$ are both finite subsets of $\mathbb{R}_+^n$ and $|B|=2$. Some $x \in co\{A\} \cap co\{B\}$ where $co$ denotes the convex hull.

1. If neither element of $B$ is in the convex hull of $A$, then is it true that some element of $A$ must be in the convex hull of $B$?

2. Say $B = \{b_1,b_2\}$ and $b_1$ is in the convex hull of $A$ but $b_2$ is not. Then is it true that some element $a$ of $A$ must be in the convex hull of $(A\backslash\{a\}) \cup \{b_2\}$?

I can see that both of these are true for $n=1$, but don't know how to check generally.

Answer 1

1) In general, no.

2) In general, no.