7-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #7 (revised) Last modified: November 7, 2007 Due Thursday, Nov. 8 at class. 1. A convex linear functional on X is a sublinear functional that meets the one additional requirement that p ( x ) 0 for all x . Prove that for arbitrary x 0 X and a > 0, the set S = { x X | p ( x - x 0 ) a } is convex. 2. A symmetric convex linear functional is a sublinear functional that meets two additional requirements: p ( x ) 0 for all x . p ( - x ) = p ( x ) for all x . (a) Prove that, if p ( x ) is a continuous symmetric convex linear functional defined on a normed linear space X , then for all x 0 X , there exists a linear functional F defined on X such that F ( x 0 ) = p ( x 0 ) and | F ( x ) | ≤ p ( x ) for all x X . It is the absolute value | F ( x ) | that makes things different. (b) Suppose that X is R 2 and p ( x ) = 2 | ξ 1 | + | ξ 2 | . Show that p ( x ) is a symmetric convex linear functional. If
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7-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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