# When is the intersection of half-space unions path-connected?

I have sets $S_1, \dots, S_m \subset\mathbb{R}^n$ ($m > n$). Each $S_j$ is the union of two half-spaces (a half-space is described by $\{x | a \cdot x \ge b \}$ for some $a \in \mathbb{R}^n$, $b \in \mathbb{R}$). I would like to know if $S_1 \cap \dots \cap S_m$ is path-connected. How can I test for this? Many thanks!

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