How many regions are created by the set of all hyperplanes defined by a set of points?
Let’s learn how many regions are created by the set of all hyperplanes defined by a set of points. The most accurate or helpful solution is served by Mathematics.
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If we have a set of points X in d-dimensional euclidean space, and we look at the set of all n-dimensional hyperplanes that are defined by any subset Y of X (in the sense of being the unique n-dimensional hyperplane containing the points of Y), ranging over all Y in P(X), is there a result for the number of regions defined by this hyperplane arrangement? edit: I just realized, I'm not sure this makes sense for arbitrary n, only n=d-1.
Answer:
I will recast the question as how many pieces can $d$ dimensional space be cut into by $n$ $d-1$ dimensional...
user114766 at Mathematics Mark as irrelevant Undo
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Answer:
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