Prove that intersection of a convex set is an convex set
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Theorem: Given any collection of convex sets (finite, countable or uncountable), their intersection is itself a convex set.
Proof: If the intersection is empty, or consists of a single point, the theorem is true by definition.
Otherwise, take any two points A, B in the intersection. The line AB joining these points must also lie wholly within each set in the collection, hence must lie wholly within their intersection.
Proof: If the intersection is empty, or consists of a single point, the theorem is true by definition.
Otherwise, take any two points A, B in the intersection. The line AB joining these points must also lie wholly within each set in the collection, hence must lie wholly within their intersection.
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3 Prove that the intersection of two convex sets is a convex set. ... We want to show that A ∩ B is also convex. Take x1,x2 ∈ A ∩ B, and let x lie on the line segment between these two points. Then x ∈ A because A is convex, and similarly, x ∈ B because B is convex.
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