Let $B_1$ and $B_2$ be two closed ball with positive radius in $\mathbb{R}^d$. We know that if $\hat{f}$ is supported in $B_1$ then f cannot be supported in $B_2$.
Do we have furthermore that there exists $C>0$ (only depending on $B_1$ and $B_2$)
$\underset{f\in L^2(\mathbb{R}^d), \|f\|_2=1,\, \text{supp} \hat{f}\subset B_1}{\inf} \|f\|_{L^2(\mathbb{R^d}\setminus B_2)}\geq C$
The result might also hold for more general sets.