I'm looking for the proof of the following/reference to such proof. At the end of the day, my goal is to confirm if this inequality holds.
Let $H_{n-1}$ be the $n-1$ dimensional Hausdorff measure and $\Omega\subseteq \mathbb{R}^n$ be a bounded domain. Then $$H_{n-1}(\partial \Omega) \geq nV_n^{1/n}\lambda_n(\Omega)^{\frac{n-1}{n}}, $$ $\lambda_n$ being the $n$ dimensional Lebesgue measure, and $V_n$ being the volume of unit ball in $\mathbb{R}^n$.
I've searched for this in Federer's Geometric measure theory, the statement 3.2.44 says that this holds for domains with rectifiable boundary. It also says that its generalized in 4.5.9 (31). I'm unable to parse the statement of theorem 4.5.9 however.