Macroeconomics Exam Review 169

# Macroeconomics Exam Review 169 - c 2001 Michael Carter All...

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0 1234 1 2 3 4 conv ? 2 P(x) conv ? 1 (0, 2.5) (.5, 2) Figure 3.3: Illustrating the proof of the Shapley Folkman theorem. 3. Every extreme point of conv ? ? is an element of ? ? . 3.211 See Figure 3.3. 3.212 Let { ? 1 ,? 2 ,...,? ± } be a collection of nonempty subsets of an ± -dimensional linear space and let x conv ± ? =1 ? ? = ± ? =1 conv ? ? . That is, there exists x ? conv ? ? such that x = ± ? =1 x ? . By Carath´ eodory’s theorem, there exists for every x ? a ±nite number of points x ? 1 , x ? 2 ,..., x ? such that x ? conv { x ? 1 , x ? 2 x ? } . For every ² =1 , 2 ,...,³ ,let ˜ ? ? = { x : ´ , 2 ,...,µ ? } Then x = ± ± ? =1 x ? , x ? conv ˜ ? ? That is, x conv ˜ ? ? =conv ˜ ? ? . Moreover, the sets ? ? are compact (in fact ±nite). By the previous exercise, there exists ³ points z ? ˜ ? ? such that x = ± ± ? =1 z ? , z ? conv ˜ ? ? and moreover z ? ˜ ? ? ? ? for at least ³ ± indices ² . 3.213 Let ? be a closed convex set in a normed linear space. Clearly,
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