Is There a Better Way to Parametrize a Special Region in n-Dimensional Space?

[Pere Callahan]

Hi,
I am looking for a "good" way to parametrize the region of \mathbb{R}^n where one coordinate, say x_1 is greater than all the others.

I came up with a possiblity to do that in hyperspherical coordinates \{r,\varphi_1,\varphi_2,\dots ,\varphi_{n-1}\}

where

0\leq r \leq\infty
0\leq \varphi_{n-1} \leq 2\pi
0\leq \varphi_\nu \leq\pi \quad\quad 1\leq\nu\leq n-2

Then for example, if I wanted to integrate over the region of \mathbb{R}^n where
x_1 \geq x_2 \dots \geq x_n
I could do it like this

\int_0^\infty dr\int_{-\frac{3}{4}\pi}^{\frac{\pi}{4}}d\varphi_n \int_{0}^{\frac{\pi}{2}-ArcTan[Cos[\varphi_n]]}d\varphi_{n-1}\dots\int_{0}^{\frac{\pi}{2}-ArcTan[Cos[\varphi_2]]}d\varphi_1 r^{n-1}Sin[\varphi_1]^{n-2}\dots Sin[\varphi_{n-1}] + + \int_0^\infty dr\int_{-\frac{\pi}{4}}^{\frac{5}{4}\pi}d\varphi_n \int_{0}^{\frac{\pi}{2}-ArcTan[Sin[\varphi_n]]}d\varphi_{n-1}\dots\int_{0}^{\frac{\pi}{2}-ArcTan[Cos[\varphi_2]]}d\varphi_1 r^{n-1}Sin[\varphi_1]^{n-2}\dots Sin[\varphi_{n-1}]


If I then sum over all permutations of \{x_2,\dots ,x_n\} I can integrate over the region where x_1 is greater than all the other coordinates. However the integration limits are somewhat unwieldy so my question is if anybody knows of a better way to parametrize the region I am interested in.

Thanks

Cheers,
Pere

 

[fresh_42]

If $x_1 > x_i$ for all $i>1$ then $y_i=x_1-x_i$ looks as a good coordinate system. This way you get simply a Cartesian quadrant.