Let $\Omega\subset \mathbb{R}^n$ be a bounded connected domain. Introduce the notation$$\Omega_r=\left\{x\in\Omega|d(x,\partial\Omega)<\frac{1}{r}\right\},\quad r>0,$$where $d(x,\partial\Omega)$ is the distance of the point $x\in\Omega$ from the boundary $\partial\Omega$. In Kroger's paper 'Estimates for sums of eigenvalues of the Laplacian' (Journal of Functional Analysis, 1994), he claimed that$$\left(\frac{r_1}{r_2}\right)^n{\rm vol}(\Omega_{r_1})\leq{\rm vol}(\Omega_{r_2})\leq{\rm vol}(\Omega_{r_1})\quad\text{for }0<r_1\leq r_2.$$Here, ${\rm vol}(\cdot)$ denotes the standard $n-$dimensional Lebesgue volume.
By the hint in that paper, I try to use the Co-Area Formula to prove the above inequalities. However, I'm confused in proving $\left(\frac{r_1}{r_2}\right)^n{\rm vol}(\Omega_{r_1})\leq{\rm vol}(\Omega_{r_2})$.
Here is my attempt:
Define $f:\mathbb{R}^n\to\mathbb{R}^n$ by $f(x)=\frac{r_1}{r_2}x$. By the Co-Area Formula, we know$$\left(\frac{r_1}{r_2}\right)^n{\rm vol}(\Omega_{r_1})={\rm vol}(f(\Omega_{r_1})).$$Therefore, we need to prove that ${\rm vol}(f(\Omega_{r_1}))\leq {\rm vol}(\Omega_{r_2})$. However, it seems that there doesn't exist any relationship between $f(\Omega_{r_1})$ and $\Omega_{r_2}$. Since $\Omega$ is only bounded and connected, for $x\in \Omega_{r_1}$, we can not figure it out that whether $d(f(x),\partial\Omega)$ is increasing or decreasing. So I can not find a way to prove the inequality presented above.
Any help will be appreciated a lot!