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Consider the following:

1) How many connected regions can $n$ hyperplanes form in $\mathbb R^d$?

2) What if the set of hyperplanes are homogeneous?

3) 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?

Q3 is what I'm really after.

Any ideas?

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  • $\begingroup$ What do you mean by “hyperplanes are homogeneous”? Are you considering hyperplanes cutting a projective space into regions? If you are really after Q3, what made you include Q2 in your post? Do you think answering that will help with Q3? If so, explaining that might be useful. $\endgroup$
    – MvG
    Jan 2, 2017 at 15:34

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