Theorem 1.The intersection of a finite number of convex sets is also convex.Proof.1. LetA,Bbe two convex sets.Take any pair of elementsp,qsuch thatp, q∈A∩B. SinceAis 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∩Bis convex.2. We prove by induction.(a) For any two convex sets, their intersection is convex. (by 1)
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