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Unformatted text preview: 1.244 may be empty, in which case it is trivially convex. Otherwise, let x . For every x x x which implies x ( x ) Therefore ( x ). Conversely, by transitivity x x y for every y for every x ( x ) which implies ( x ) . Therefore, = ( x ) which is convex. 1.245 To show that is strictly convex, assume that x , y are such d ( x ) = d ( y ) with x = y . Suppose d ( x ) = ( ( 1 , x ) , ( 2 , x ) ..., ( 2 , x ) ) In the order 1 , 2 ,..., 2 , let be the first coalition for which ( , x ) = ( , y ). That is ( , x ) = ( , y ) for every < (1.32) Since ( , x ) = ( , y ) and d ( x ) is listed in descending order, we must have ( , x ) > (...
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- Fall '10