Consider 2 parallel hyperplanes of the type
$0 \leq \langle w,x \rangle + b$ and $\langle w,x \rangle + b < c$ where $x, w \in \mathbb{R}^n ; b,c \in \mathbb{R}$
cutting an $n$-ball with radius $r$ and centered at origin, $||x||_2 \leq r$.
Hence, the volume of intersection of the 2 hyperplanes forms a spherical segment.
What is the volume of the intersection in closed form? (or an upper-bound for that matter)
Is it possible to get a closed form using sampling? If yes, can I have some guidance please?
Is there some other way to look at this rather than a spherical segment? probably as the difference in volumes of spherical caps?