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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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