I want to show that $$\frac{1}{\prod\limits_{i=0}^{i=n}A_{i}}=n!\int\limits_{\mid \Delta^{n}\mid}\frac{d\sigma}{\left( \sum s_i A_i \right)^n}$$
Where $d\sigma$ is the lebesgue measure on the standard n-simplex $\mid \Delta^{n}\mid$ $s_i$ are dummy integration variables. One should take $\sum s_{i}=1$ where the s represents a coordinate system on the n-simplex.
One can proceed by induction. $n=1$ $n=2$ are easy. one assumes that the above formula holds for $n-1$ and try to prove it for in. I would like to use stokes theorem for proving this formula. One takes $d\sigma$ to be the volume form "n-form" on the oriented n-simplex.