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Unformatted text preview: 4.45 Assume not. That is, assume that y = ( x 1 ) ( x 2 ) conv Then by the (strong) separating hyperplane theorem (Proposition 3.14) there exists a linear functional on such that ( y ) > ( a ) for every a (4.48) where ( ) = ( ( x 1 ) ( x 2 )) = ( ( x 1 )) ( ( x 2 )) is a functional on . By the mean value theorem (Theorem 4.1), there exists some x [ x 1 , x 2 ] such that ( x 1 ) ( x 2 )) = ( )[ x ]( x 1 x 2 ) = [ x ]( x x 2 ) = ( ) for some contradicting (4.44). 4.46 Define : [ , ] by ( ) = ( ( ) ( ) ) ( ) ( ( ) ( ) ) ( ) [ , ] and is differentiable on , ) with ( ) = ( ( ) ( ) ) ( ) (...
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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.
- Fall '10