Macroeconomics Exam Review 47

Macroeconomics Exam Review 47 - c 2001 Michael Carter All...

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Let y be any element in the unit ball 𝐵 . Then y 𝐵 and x 1 = x + 𝑟 y 𝑆 x 2 = x 𝑟 y 𝑆 so that x = 1 2 x 1 + 1 2 x 2 x is not an extreme point. We have shown that no interior point is an extreme point; hence every extreme point must be a boundary point. 1.221 We showed in Exercise 1.220 that ext( 𝑆 ) b( 𝑆 ). To show the converse, assume that x is a boundary point which is not an extreme point. That is, there exist x 1 , x 2 𝑆 such that x = 𝛼 x 1 + (1 𝛼 ) x 2 0 < 𝛼 < 1 This contradicts the assumption that 𝑆 is strictly convex. 1.222 If 𝑆 is open, int 𝑆 = 𝑆 . Since 𝑆 is convex 𝛼 x + (1 𝛼 ) y 𝑆 = int 𝑆 for every 0 𝛼 1 A fortiori for every x = y 𝛼 x + (1 𝛼 ) y 𝑆 = int 𝑆 for every 0 < 𝛼 < 1 𝑆 is strictly convex. 1.223 Let 𝑆 be open and
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