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I found this formula for calculating maximum number of regions created by r hyper-planes in n-dimensional space (n<=r)

                      rC0 + rC1 + ....... + rCn

How can this formula be proved??

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  • $\begingroup$ The number of regions depends on the particular hyperplanes. I think your formula is for generic hyperplanes. $\endgroup$
    – Sam Hopkins
    Commented Mar 22 at 1:24
  • $\begingroup$ @SamHopkins Yes, I guess it's for the maximum number of regions that any r hyperplanes can make in n-dimensional space $\endgroup$
    – Mazen Saaed
    Commented Mar 22 at 1:27
  • $\begingroup$ Please use TeX on this site. $\endgroup$
    – GH from MO
    Commented Mar 22 at 3:00
  • $\begingroup$ These should be some type of cake numbers. There is also a result what happens if some of them are parallel, see the works by Wetzel. Also: jlmartin.ku.edu/MiniCollege2012/slides.pdf $\endgroup$
    – Per Alexandersson
    Commented Mar 22 at 11:49

1 Answer 1

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One proof is in my book Enumerative Combinatorics, vol. 1, second ed., Proposition 3.11.8. The first proof is due to L. Schläfli, written in 1850-52 but not published until 1901 in Neue allgemeinen schweizerischen Gesellschaft für die gesamten Naturwissenschaften, vol. 38, IV, Zurich, 1901.

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