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PR1 - Theorem 1 The intersection of a nite number of convex...

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Theorem 1. The intersection of a finite number of convex sets is also convex. Proof. 1. Let A , B be two convex sets. Take any pair of elements p , q such that p, q A B . Since A is convex, by definition, αp + (1 - α ) q A (0 α 1) . Similarily, αp + (1 - α ) q B (0 α 1) . Then the following is true: αp + (1 - α ) q A B (0 α 1) . Therefore, by definition, A B is convex. 2. We prove by induction. (a) For any two convex sets, their intersection is convex. (by 1)
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