Answer by zeb for On the extreme points of two convex sets
No: consider the line segments $\{0\}\times[-1,1]$ and $[-1,1]\times\{0\}$ in $\mathbb{R}^2$.
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Let $A$ and $B$ be two compact convex sets (which may be assumed to be polytopes) in $\mathbb R^n$ such that $A\cap B\ne\emptyset$. Is it then always true that either $A\cap\text{ext}B\ne\emptyset$ or...
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