PR1 - Theorem 1. The intersection of a finite number of...

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Unformatted text preview: 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 Similarily, αp + (1 − α)q ∈ B Then the following is true: αp + (1 − α)q ∈ A ∩ B Therefore, by definition, A ∩ B is convex. 2. We prove by induction. (a) For any two convex sets, their intersection is convex. (by 1) (b) Assume that the intersection of n convex sets B = A1 ∩ A2 ∩ · · · ∩ An is convex. We want to prove that the intersection of n + 1 convex sets C = A1 ∩ A2 ∩ · · · ∩ An ∩ An+1 is convex. Then we note that C = B ∩ An+1 . By above assumption, B is convex. Therefore C is also convex by (a). (0 ≤ α ≤ 1). (0 ≤ α ≤ 1). (0 ≤ α ≤ 1). 1 ...
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This note was uploaded on 02/04/2010 for the course COMPUTER S cs300 taught by Professor Unkown during the Spring '08 term at Korea Advanced Institute of Science and Technology.

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