Consider a finite hyperplane arrangement $\mathcal{A}$ over $\mathbb{R}^n$. Let the regions given by $\mathcal{A}$ be $\mathcal{R}(\mathcal{A})=\{A_1,\dots A_m\}$ for some $m$.

For any index set $I\subseteq \{1,\dots,m\}$, is it true that $$ B(I)=\left(\bigcap_{i\in I} \overline{A_i}\right) \backslash \left(\bigcup_{i\not\in I}\overline{A_i}\right), $$ is convex?

It is easy to see that $\bigcap_{i\in I}\overline{A_i}$ is convex. But I haven't found a way to show that $B(I)$ is convex. I have to say that I am rather unexperienced in geometry...