I'm currently reading a text about hyperplane arrangements in a euclidian vector space V. The text says:
[...] without loss of generality, we may assume that some of the hyperplanes intersect in a single point otherwise we may pass to a cross region.
Would someone be so kind and elaborate why we may assume that some of the hyperplanes intersect in a single point?
The only possibility that came to my mind so far was, for example, three hyperplanes in $\mathbb{R}^3$ intersecting in a single point, but i'm quite confident i'm still missing a few things here.
For context: this is about hyperplane arrangements that partition the euclidian vector space $V$ into chambers and i'm interested in the reflection group generated by the reflections in the walls of the hyperplane arrangement.
Suppose you arrangement is $A=\{H_i: i\in I\}$, where $H_i=\{f_i(x)+a_i=0\}$, where each $f_i$ is a nonzero linear functional on your vector space $V$. Let $F=\{f_i: i\in I\}$ denote the corresponding set of linear functionals. It is a nice linear algebra exercise to prove that the intersection $K$ of kernels of functionals in $F$ is nonzero if and only if for every subset $J\subset I$, $$ \bigcap_{i\in J} H_i $$ is not a singleton (i.e. is empty or has positive dimension). Thus, either you have your point intersection, or $K\ne \{0\}$. In the latter case, $V\cong K\oplus V'$ and, accordingly, your arrangement consists of hyperplanes $H_i=K\oplus H_i'$, where $H_i'$ are affine hyperplanes in $V'$. The associated arrangement $A'= \{H'_i: i\in I\}$ now has the property that a subcollection of hyperplanes has point-intersection. This is what they mean by "pass to a cross region" (which is quite sloppy).