I have to calculate the following integral:
$$ \int_{x_1=-1}^{1} \int_{y_1=-1}^{1} \cdots \int_{x_N=-1}^{1} \int_{y_N=-1}^{1}\left( \prod_{i=1}^{N} \prod_{j =1, \neq i}^N \frac{1}{|\vec{r_i}-\vec{r_j}|^{\frac{1}{N-1}}} \right) \delta({\sum_{i=1}^N |\vec{r_i}|^2 - 1}) d^2r_1 d^2r_2 \cdots d^2r_N $$
where $N$ is a large number, $\delta$ is a delta Dirac function that gives us the condition, $\vec{r_i}$ is a 2-dimensional cartesian vector with components $x_i$ and $y_i$, and $|\vec{r_i}-\vec{r_j}|$ means the norm of vector $\vec{r_i}-\vec{r_j}$ and here the power is $\frac{1}{N-1}$. $i \neq j $ so this integral is not divergent. we also consider a cut-off for the norm $|\vec{r_i}-\vec{r_j}|$, that is $|\vec{r_i}-\vec{r_j}| \geq d$ that $d$ is a tiny value and makes the integral not be divergent. I think I can write the above integral on the surface of a $2N$-dimensional unit ball just by renaming, but I don't know if it helps me or not! thank you for your help, regards.