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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