I want to show that
$$\frac{1}{\displaystyle\prod_{i=0}^{i=n}A_{i}}=n!\int\limits_{|\Delta^{n}|}\frac{\mathrm d\sigma}{\left( \displaystyle \sum\limits_i s_i A_i \right)^n}$$
where $\mathrm d\sigma$ is the Lebesgue measure on the standard $n$-simplex $|\Delta^{n}|$, and $s_i$ are dummy integration variables. One should take $\sum_i s_{i}=1$ where $s$ represents a coordinate system on the $n$-simplex.
One can proceed by induction. $n=1$ $n=2$ are easy. One then assumes that the above formula holds for $n-1$ and tries to prove it for $n$. I would like to use Stokes theorem for proving this formula. One takes $\mathrm d\sigma$ to be the volume form “$n$-form” on the oriented $n$-simplex.
