# Number of Regions for a Central Hyperplane Arrangement

This question has likely been answered in full detail before, so any references would be greatly beneficial. The question I have is as follows:

Suppose we have $$m$$ central hyperplanes in $$\mathbb{R}^n$$, that is, $$m$$ hyperplanes of dimension $$n-1$$ that cross through the origin. How many regions does this arrangement split $$\mathbb{R}^n$$ into?

It isn't difficult to find the answer for noncentral arrangements, this is given by Zaslavsky's Theorem. Furthermore there is a detailed treatment of hyperplane arrangements in "An Introduction to Hyperplane Arrangements" at https://www.cis.upenn.edu/~cis610/sp06stanley.pdf. However, I wasn't able to find the result I am looking for.

What I could find is that if we allow the hyperplanes to be in general position, then the arrangement $$A$$ of $$m$$ hyperplanes in $$\mathbb{R}^n$$ has $$\begin{equation} r(A) = \sum_{j=0}^n {m \choose j} \end{equation}$$ regions.

For the central arrangement question, the interesting case is when $$m > n$$. I believe that when $$m \leq n$$, the number of regions is simply $$2^m$$.

When $$n\ge 1$$ and $$m\ge 1$$, the answer is $$2\sum_{j=0}^{n-1}\binom{m-1}j.$$To derive this from the non-central answer, given $$m$$ central hyperplanes in $$\mathbb R^n$$, let $$P$$ be one of the planes. Define two planes $$P^+$$ and $$P^-$$ which are parallel to $$P$$, such that the origin is between $$P^+$$ and $$P^-$$. Every region is on one side of $$P$$ or the other, so it either passes through $$P^+$$ or $$P^-$$. The plane $$P^+$$ is $$(n-1)$$ dimensional, so it is divided by the other $$m-1$$ planes into $$\sum_{j=0}^{n-1}\binom{m-1}j$$ regions, by the previous result. These regions of $$P^+$$ correspond exactly to the original regions which are on the $$P^+$$ side. You then multiply by $$2$$ to also account for the $$P^-$$ regions.