how to show the following properties are holds in $\mathbb{R}^n$? for two bounded set $\Omega_1$, $\Omega_2 \subset \mathbb{R}^n$ and $\Omega_1$, $\Omega_2 \subset B_R$, $B_R$ is the Ball radius $R$ and centered at origion, then
prop: $|\Omega_1 \cap \Omega_2| |x_{\Omega_1} - x_{\Omega_2}| \le C(R) |\Omega_1 \Delta \Omega_2|$, where $x_{\Omega_i}$ is mass center of $\Omega_i$ for $i = 1,2$, $\Omega_1 \Delta \Omega_2 = (\Omega_1 \setminus \Omega_2) \cup (\Omega_2 \setminus \Omega_1)$. $|\Omega|$ is volume of $\Omega$, $C(R)$ is a constant that only depends on $R$.
I consider if $\Omega_1 = \Omega_2$, the left is $0$, so it seems $|x_{\Omega_1} - x_{\Omega_2}|$ is essentially used, but I can't figured out how to engaged that.
This question is come from paper Faber-Krahn inequalities in sharp quantitative form page 11.
