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Let $x_1$ and $x_2$ be two vectors in $\mathbb{R}^2$. Consider an objective function $V(x)=(\|x_1\|^2-d_1^2)^2 + (\|x_2\|^2-d_2^2)^2$ where $d_1$ and $d_2$ are two positive constants, and the vector $x=[x_1^T,x_2^T]^T\in\mathbb{R}^4$. Define the sublevel set $\Omega_c=\{x\in\mathbb{R}^4 \ |\ V(x)\le c\}$ Can anyone show how to prove this set $\Omega_c$ is compact (closed and bounded)? Thanks.

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Hint: For closedness check continuity of $V$, for boundedness, check what happens to $V(x)$ for $\|x\|\to\infty$.

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    Inverse images of sets always exist: The inverse image of $A$ under $V$ is the set $\{x\ :\ V(x) \in A\}$.2012-03-26