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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
There are a few facts that may help you (although, I believe you need to rephrase your question to make it more clear). First, if $P$ is affine, i.e., $P(x)=Ax+b$, and $G$ is convex, then $F$ is convex. Second, assume that $P$ is given by $P(x)=(P_1(x),\ldots, P_n(x))$ and each $P_i:\mathbb{R}\to\mathbb{R}$ is convex. Assume that $G(z_1,\ldots, z_n)$ is nondecreasing in each $z_i$ (when other variables are fixed) for each $i$. Then $F$ is convex. – Pantelis Sopasakis Nov 2 '14 at 14:22

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