Ami Pro - LN040811

Ami Pro - LN040811 - C 1 C 2 is convex C 1 UCSD Economics...

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Lecture Notes, April 8, 2011 Convexity A set of points S in is said to be convex if the line segment between any two points R N of the set is completely included in the set. S is convex if x, y S, implies . z z   x   1   y ,0    1   S S is said to be strictly convex if x, , implies y S , x y ,0    1 . x   1  y interior S The notion of convexity is that a set is convex if it is connected, has no holes on the inside and no indentations on the boundary. A set is strictly convex if it is convex and has a continuous strict curvature (no flat segments) on the boundary. Economically, this notion corresponds to "diminishing marginal utility" "diminishing marginal rate of substitution" "diminishing marginal product" . Properties of Convex Sets Let be convex subsets of R N . Then: C 1 , C 2 is convex, C 1 C 2 is convex,
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Unformatted text preview: C 1 C 2 is convex C 1 UCSD Economics 113 Spring 2011 Ms. Stephanie Fried Prof. R. Starr 1 CB046/Starr LNBrouwer April 6, 2011 11:2 1 The unit simplex in R N , is P = b p | p R N , p i , i = 1 , . . ., N, N s i =1 p i = 1 B . (5.1) The unit simplex is a (generalized) triangle in N-space. Note that P is compact (closed and bounded) and convex. Theorem 5.1 (Brouwer Fixed-Point Theorem) Let f ( ) be a continuous func-tion, f : P P . Then there is x * P so that f ( x * ) = x * . The four properties assumed in the Brouwer Fixed Point Theorem con-tinuity of f , closedness, boundedness, and convexity of P are all essential to the theorem. Omit any one of them and the theorem fails. The result can be generalized from P to any compact convex set....
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Ami Pro - LN040811 - C 1 C 2 is convex C 1 UCSD Economics...

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