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Let $U$ be a bounded open convex set of $\mathbb{R}^d$, and $\Phi$ a differentiable map from $U\times \mathbb{R}$ to $\mathbb{R}^d$. The object of interest for me is $R=\Phi(U\times \mathbb{R})$. The derivative of $\Phi$ can be written $$ d\Phi(C,z).(h,s)=f(C,z)h-2zs\Phi(C,z) $$ where $f(C,z)>0$ everywhere. How can I obtain information about the boundary of $R$? How could I prove for instance that $R$ is simply connex? What other information could I get?

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