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Halfplanes cover


Suppose that the plane can be covered by n half-planes. Then we can find three of these half-planes which cover the plane.

Proof: Suppose A_i are the given half-planes and B_i are their complements. The condition \displaystyle \bigcup A_i = \mathcal{P} is equivalent to \displaystyle \bigcap B_i=\emptyset. If no three half-planes cover the plane then every three of the B_i intersect. Since B_i are convex sets, it follows from Helly’s theorem that their intersection is not void. Contradiction.
Therefore, we can find three of the given half-planes which cover \mathcal{P}.

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