If $K'$ is the compact support of a $ℝ$-valued function on an open set $Ω⊆ℝ^d$, can we find a closed ball $K$ with $K⊆Ω$ and $K'⊆K$?

Question

Let

• $d\in\mathbb N$
• $\Omega\subseteq\mathbb R^d$ be non-empty and open
• $\phi:\Omega\to\mathbb R$ be continuous with compact support $\operatorname{supp}\phi$

Can we find a closed ball $K$ such that $K\subseteq\Omega$ and $K':=\operatorname{supp}\phi\subset K$?

Answer 1

Take $\Omega = B(0,2) \backslash \bar{B}(0,1)$, and $\phi(x) = \|x\| - 3/2$.

Then $K' = \{ x \in \Omega | \|x\| = 3/2 \}$, and there is no ball that contain $K'$ that is inclued in $\Omega$