I was trying to think of a nice proof that a set $S$ of $n$ hyperplanes in general position in $\mathbb{R}^d$, divide the space into $F(n,d)$ regions, where \begin{equation} F(n,d) = \sum_{i=1}^d \binom{n}{i} \end{equation}
I've seen the different sets of notes about this problem around the web (in particular, Richard Stanley's), but they are all so general, and to me, very complicated.
The idea for a proof I had was as follows. First, order the dimensions of $\mathbb{R}^d$ as $x_1, x_2, \ldots,x_d$. Now place the hyperplanes in a "hypercube" that is dimension-aligned, and is big enough that it doesn't add any regions (that is, the number of regions inside the cube is the same as if the cube wasn't there).
Now put the lexicographic order on the points in the hypercube. Then every region in the cube, as a closed bounded set, has a unique "smallest" point. This point is the intersection of $d$ hyperplanes, but some of those might be sides of the bounding cube. That is, each such point can be written as the intersection of $d$ hyperplanes, with at most $d$ coming from our original set $S$. Also note that if $k$ of the hyperplanes are from our original set, the remaining $d-k$ are uniquely determined as sides of our cube: the ordering on points induces an ordering on sides, and we always pick the smallest $d-k$ sides still intersecting our $k$ hyperplanes.
We then have a map $\{$regions$\}\rightarrow\{$smallest point$\}\rightarrow\{$subsets of $S$ of size at most $d\}$
That last entry has size $F(n,d)$. We can see the count of regions is exactly this by the following facts:
- no two subsets of $S$ have the same intersection: this follows from general position.
- no two regions have the same smallest point: if they did, they couldn't have the same elements of $S$ determining that point (since regions are disjoint), so we would have a contradiction with the previous point.
- every point $p$ that comes from intersecting $d$ hyperplanes is a "smallest point" for some region: just take the region defined by these $d$ hyperplanes and the set of points bigger than $p$.
So the two arrows above are bijective.
Here is a picture of this for 4 lines in $\mathbb{R}^2$, with the bounding box drawn. Red points label the "smallest" points for each region, with small arrows added to indicate which region is associated to which point.
Finally, my questions:
- Is there anything wrong with this proof?
- Are there other proofs out there like this one?
- Can this argument be generalized to give an upper bound? That is, if $S$ is any set of $n$ hyperplanes (not necessarily in general position), than the number of regions is $\le F(n,d)$?
(3) isn't obvious to me: I don't know what to do when multiple regions share the same smallest point.