Macroeconomics Exam Review 171

Macroeconomics Exam Review 171 - polyhedron = polytope...

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Unformatted text preview: polyhedron = polytope Conversely, assume is a nonempty compact polyhedral set in a normed linear space. Then, there exist linear functionals 1 , 2 ,..., in and numbers 1 , 2 ,..., such that = { x : ( x ) , = 1 , 2 ,..., } . We show that has a finite number of extreme points. Let denote the dimension of . If = 1, is either a single point or closed line segment (since is compact), and therefore has a finite number of extreme points (that is, 1 or 2). Now assume that every compact polyhedral set of dimension 1 has a finite number of extreme points. Let , = = 1 , 2 ,..., denote the hyperplanes associated with the linear functionals defining (Exercise 3.49). Let x be an extreme point of . Then is a boundary point of (Exercise 1.220) and therefore belongs to some . We claim that x is also an extreme point of the set...
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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