Let $A,B\subseteq \mathbb{R}^{n}$ be compact sets, with $A\cap B\neq \emptyset$, such that $A\cup B$ is convex. Also let $L=\{\lambda(t)=(1-t)x+ty \, : \, 0\leq t\leq 1\}$ for $x\in A$ and $y\in B$. Is it true that $$L\cap (A\cap B)\neq \emptyset \, ?$$This seems fairly intuitive when yout think about cubes in $\mathbb{R}^{3}$, yet I wasn't able to prove it. The first thing that came to mind was to assume $L\cap (A\cap B)= \emptyset$ and then construct (using the fact that $L$ is connected and intersects $A$, $A-B$, $B$ and $B-A$) a descending sequence of compact intervals $I_{n}\subseteq [0,1]$, with $\mathrm{diam} I_{n}\to 0$, so that $\bigcap I_{n}=\{p\}$ and $\lambda(p)\subseteq L$, and finally try to argue that it must be $\lambda(p)\in A\cap B$. I didn't go through with it because I couldn't find a sequence fulfilling $\mathrm{diam} I_{n}\to 0$. I wonder if anyone can give a different insight on how to approach this problem.
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