I am trying to understand a line of the proof of part I-A of the Sobolev Embedding Theorem in Sobolev Spaces by Adams (section 4.16, page 89). Specifically, the following value integral is presented
$$\int_{C_{x,\rho}} |x - y|^{(m-n)p'} dy$$
where $C_{x,\rho}$ is a cone in $\mathbb{R}^n$. The next line of the proof states that because $(m - n)p' > -n$, the integral is finite. I don't see how this follows. In particular, if $n = 2$, couldn't $(m-n)p' = -1$ so $|x - y|$ would be infinite for $x = y$?
There is a hypothesis of the Sobolev Imbedding Theorem relating $m$, $n$, and $p$ that you are neglecting to take into account.