Let $|\Sigma|$ be the underlying set of some fan $\Sigma$ in $\mathbb{R}^n$. It is well known that in general there is no coarsest fan structure on $|\Sigma|$. However, there may be some special rays that occur as cones in any fan structure on $|\Sigma|$, for example extremal rays. Is there a characterization of these rays? In particular, have these rays been studied in the context of tropical geometry?
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