TITLE: How to compute the volume of intersection between two hyperspheres
QUESTION [12 upvotes]: Let's say I have two n-spheres and I've no prior knowledge about the spheres (such as one of the sphere might be inside the other one) and I need to compute the volume of the intersection of the two hyper-spheres. Is there an efficient and generic way to compute the hyper-spherical cap of the intersection of these n-spheres? Note that the hyperspheres are expected to be very high dimensional such as 4096.

REPLY [2 votes]: Following the idea of using caps, The Wikipedia article has a mention of a nice asymptotic result: If $h$ is the height of the cap and $r$ is the radius of the hypersphere, then $V^{cap}_n\rightarrow V_n(1-F((1-h/r)\sqrt{n}))$ for large $n$, where $F$ is the integral of the standard gaussian.
In our case $h=\frac{2r-d}{2}$ and we are interested in the intersection which is $2V_n^{cap}$, so we get $2V^{cap}_n\approx V_n\,2(1-F(\frac{d}{2r}\sqrt{n}))$.
This shows that the intersection, even normalized by $V_n$ goes to $0$ as the dimension gets large. Pretty cool.