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Suppose we have $F: R^n \longrightarrow R$ , $P: R^n \longrightarrow R^n$ and $G: R^n \longrightarrow R$ all nice- let's say given by polynomial and $P$ is invertible - such that $F(x) =G( P(x) )$. Is it possible to relate the conditions of convexity of $F$ to conditions on $G$. For example if $X$ is the image of $R^n$ under $P$, i.e. $X =P(R^n)$. Then $X$ is a semi algebraic subset of $R^n$. Now does $G$ being convex on $X$ imply $F: R^n \longrightarrow R$ is convex? If this is not the case in general are there conditions on $P$ which help here?

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What does "being convex on $X$" mean, when $X$ is not necessarily a convex set? –  user53153 Dec 18 '12 at 6:43
Hmm for example that the Hessian is psd at every point.. –  Benedikt Dec 21 '12 at 8:39

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