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Unformatted text preview: S and c [0 , 1] we have that (1) f ( cx + (1c ) y ) cf ( x ) + (1c ) f ( y ) (2) g ( cx + (1c ) y ) cg ( x ) + (1c ) g ( y ) Adding (1) and (2) yields h ( cx + (1c ) y ) ch ( x ) + (1c ) h ( y ) which shows that h is also a convex function. 13. Since f is convex we know that for 0 k 1 (1) f ( kx + (1k ) y ) kf ( x ) + (1k ) f ( y ). Multiplying both sides of (1) by c 0 shows that g is also a convex function. Similarly, multiplyng both sides of (1) by c 0 shows that g is a concave function....
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 Spring '07
 HOCHBAUM
 Operations Research

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