Consider the following:
1)How many connected regions can $n$ hyperplanes form in $\mathbb R^d$?
- How many connected regions can $n$ hyperplanes form in $\mathbb R^d$?
- What if the set of hyperplanes are homogeneous?
(Denote the first as $f(d,n)$ and the second as $g(d,n)$, and assume that $n>>d$)
- Given a set of $n$ pairs of hyperplanes, such that each pair is parallel, what is the maximum number of regions that can be formed?
I saw here,here and here that the answer to (1) is
$$f(d,n)=\sum_{i=0}^d {n \choose i}$$
However I find it non-trivial to generalize the proof of $\mathbb R^2$ that was provided to $\mathbb R^d$ (without using "lower"/"upper" descriptions). Is there any "nice" way to show it recursively?
Regarding 2, I have the feeling that $g(d,n)=2^d+2\cdot(n-d)$, since the first $d$ hyperplanes can disconnect it into $d$ regions, and any additional lines will go through two antipodal regions. However, I'm having troubles to show it formally. Another assumptionQ3 is that the effect of affinity results in lower the dimension by 1, i.e. $g(d,n)=f(d-1,n)$what I'm really after.
Any ideas?
(This is definitely not a homework question, just trying to cut my $d$-dimensional cake fairly among my friends. A reference will also do the trick).
