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◊. The following property of convex sets (which you are asked to prove in an exercise) is sometimes useful. We show by contradiction. Then, for every , and, thus, . U (x) = {y ∈ X : y t x} . When you save your comment, the author of the tutorial will be notified. Thus, the set is a collection of disjoint open intervals of the form , , or . f (x) has a closed graph: that is, if fxn;yng!fx;ygwith yn 2f (xn), then y 2f (x). For any l such that , for some . There is no , which according to is strictly below all members of . The main contribution of the paper is the proof that any element in the convex hull of a decomposably antichain-convex set is Pareto dominated by at least one element of that set. Since and , it must be that and, therefore, . There exists p 2Rn;p 6= 0, and c 2R such that X ˆfy jyp cg and xp

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