Take $w,z\in R^{n}$. I am interested in integrating (as generically if possible) $$\int_{\Omega}(w \cdot z)^{1-\sigma} d z$$ Where the domain of $\Omega$ is $1$ dimension, and includes the convex combinations of $z$. i.e. $\Omega \equiv \{z | \sum z = 1 \}$
Obviously, for $n=2$, this is easy to rearrange: $$\int_{0}^{1} \left(w_1 z_1 + w_2 (1-z_1)\right) d z_1$$ $$ = \frac{w_2^{2-\sigma} - w_1^{2-\sigma}}{(\sigma - 2)(w_1 - w_2)}$$
And the mathematica code is:
Integrate[(w[1] z[1] + w[2] (1 - z[1]))^(1 - σ), {z[1], 0, 1}] // FullSimplify
But it is a little harder to cheat like this as $n$ gets larger, and I am hoping for more general patterns. How would I set up the simple $n=2$ example using the convex combination directly? I tried things like:
Integrate[(w[1] z[1] + w[2] z[2])^(1 - σ), {z[1], 0, 1}, {z[2], 1 - z[1], 1 - z[1]}] // FullSimplify
Integration over a convex combination of a region: $\int_{\Omega} (w_1 z_1 + w_2 z_2)^{1-\sigma} d (z_1, z_2)$ where $\Omega = \{ z_1 + z_2 = 1\}$
...but I couldn't get them to work.