We say that a function f R n R is convex (where R) if for any x, y R n and 0, 1 , we have f(x (1 )y) f(x) (1 )f(y) (1 ) 2 x y 2 2 Note that the usual notion of convexity is the same as 0 convexity (a) (10pts) Given R, show that the following statements are equivalent (1) The function f is convex (2) The function x 7 f(x) 2 x 2 2 is convex (b) (10pts) Is the function R n x 7 x1 convex for some 0 Justify your

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We say that a function f   R n R is  convex (where R) if for any x, y R n and  0, 1 , we have f(x (1 )y) f(x) (1 )f(y) (1 ) 2 x y 2 2   Note that the usual notion of convexity is the same as 0 convexity  (a) (10pts)  Given R, show that the following statements are equivalent  (1) The function f is  convex  (2) The function x 7 f(x) 2 x 2 2 is convex  (b) (10pts)  Is the function R n x 7 x1  convex for some 0  Justify your

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Question: We say that a function f : R n R is -convex (where R) if for any x, y R n and [0, 1], we

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