Is there an algorithm for computing the volume of the intersection of $m$ balls in $\mathbb{R}^n$ for $m,n \in \mathbb{N}$? That is, we want to compute the volume of
$$ \bigcap_{k=1}^m \mathcal{B}(C_k,r) $$ where $r > 0$ and $C_k \in \mathbb{R}^n$ with $\|C_k\| = \|C_{k+1}\|$
I know about the existence of a polynomial randomized algorithm for approximating the volume of general convex bodies. My question is about the existence of an algorithm which takes advantage of the particular convex bodies: intersection of balls