I am reading some notes on de Giorgi's methods in the regularity of elliptic equations, and have come across a step which I can't make sense of. The claim is as follows (see Lemma 10 in the linked notes: the constant is not present there, but is necessary I think).
If $A\subset\mathbb{R}^n$ is measurable and $y\in\mathbb{R}^n$ is fixed, then $$\int_A\frac{1}{|x-y|^{n-1}}dx\leq n\omega_n^{\frac{1}{n}}|A|^{\frac{1}{n}}$$ with equality attained for a ball centered at $y$.
This is described as an isoperimetric-type inequality, but I can't for the life of me see where the perimeter comes into this. I have tried to relate it to the usual isoperimetric inequality, but it seems like the inequalities go in different directions. I also tried using the coarea formula to no avail. Any help in deciphering this would be greatly appreciated.