Special region of R}^n

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