I am trying to estimate the probability of an event in an algorithm. For simplicity, assume there are two hyperspheres of radius $r$, at a distance $r$ from each other. I am looking to see how the volume of the intersection of the hyperspheres compares to the volume of a single hypersphere as dimensionality increases.
I consulted wikipedia here: https://en.wikipedia.org/wiki/Spherical_cap And here: http://en.wikipedia.org/wiki/Volume_of_an_n-ball
And have found the ratio $V_{intersect}/V_{hypersphere}$ reduces to analyzing the asymptotic value of: $f(d) = \int_0^{\pi/3} sin^d(t)dt$.
As $d \to \infty$, $f(d)$ goes to $0$. Futhermore, it seems like this is going to be $O(\frac{1}{2^d})$, since the $d$ is in the exponent of the above integral.
This is slightly disappointing and I'm wondering if my analysis checks out. Can anyone offer their thoughts?