Hyperplane arrangement

Remarks

The arrangement is central if all its hyperplanes have a common point. Its intersection poset $L(\mathcal{A})$ is the set of all nonempty intersections of subsets of $\mathcal{A}$, including $K^d$ itself as the empty intersection, partially ordered by reverse inclusion. For $K = \R$, the regions of $\mathcal{A}$ are the connected components of $\R^d \setminus \bigcup_{H \in \mathcal{A}} H$; each is an open convex set.

Example: in $\R^2$, the three lines $x = 0$, $y = 0$ and $x + y = 1$ form an arrangement. It is not central, since the three lines have no common point. They cut the plane into $7$ regions, one of them the bounded open triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$.

Non-example: the unit circle together with the line $y = 0$ is not a hyperplane arrangement, since the circle is not a hyperplane. An infinite family such as the lines $x = k$, $k \in \Z$, is not one either, since an arrangement is finite.