I'm trying to compute a lower bound to the integral $$\int_{(x,\theta)\in S^{n-1}\times(\frac{\pi}{4},\pi - \frac{\pi}{4}) }\min\{ x_1^2, \ldots, x_n^2, \cot^2\theta \}\cdot\sin^{n+1}\theta\ dx\ d\theta,$$ in terms of $n$ and other constants. This bound should be most tight as possible, an equality would be great but I'm not counting on this.
Until now all my attempts failed, I just can't find or approximate the regions of interest in $S^{n-1}$ (what I tried is to fix $\theta$ and find the corresponding region in $S^{n-1}$ satisfying $x_i^2\leq \cot^2\theta$ for some $i$). If someone can help me, it would be wonderful. Thank you very much!
I came up with this: $$\min\{x_1^2, ..., x_n^2, \cot^2\theta\}\cdot\sin^{n+1}\theta \leq \min\{x_1^2, ..., x_n^2\} \leq \frac{\sqrt{n}}{n},$$ where I used the approximations $\sin \theta \leq 1, \cot^2\theta \leq 1$ and in the last inequality I took the worst case, which would be $x_1 = x_2 = \dots = x_n$, now that if there is an $x_i > 1/n$, then there is another $x_j < 1/n$ and that would be the minimum. So using $\sqrt{n \cdot x_i^2} = 1$, you get $x_i = \frac{\sqrt{n}}{n}$. Probably there is a better approximation taking more care of the trigonometric functions, but this is all I could come up with.