Sorry for asking a basic question but this did not get answered on M.SE.


Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that 
$$\int\_{\partial\Omega} \frac{1}{|y|^{n-2}} dS(y) < \infty$$
where $dS$ is the surface measure.


My problem is that there is no easy way (for me) to convert this integral via chart maps etc because it's a Lipschitz domain (and not a graph which would be easy). If the integral were over an open subset of $\mathbb{R}^d$ (i.e. a normal integral) then it's easy to show using polar coordinates.



How to show this integral on boundary of Lipschitz domain is finite?
The integral over the complement of any nbd of the origin is finite. You can bound the contribution to the integral in a nbd of the origin, changing variable with a bi-lipschitz local chart: This reduces to the case of a flat boundary, that gives a finite value. Note that the area formula is not even needed, but only the elementary inequality.


