Surface of the intersection of $n$ balls Suppose there are $n$ balls (possibly, of different sizes) in $\mathbb R^3$ such that their intersection $\mathfrak C$ is non-empty and has a positive volume (i.e. is not a single point). Apparently, $\mathfrak C$ is a convex body with a piecewise smooth surface — a "quilt" of sphere fragments. Let $f(n)$ be the maximal number of fragments that can be achieved for a given $n$.
Is there a simple formula or recurrence relation for $f(n)$?

Surface and volumes can be calculated analytically for any n value. For intersections of n=5 spheres and more, they can be calculated from the 4 by 4 intersections: see theorems 4.5 and 4.6 in my book chapter:
Spheres Unions and Intersections and some of their Applications in Molecular Modeling, In: Distance Geometry: Theory, Methods, and Applications, chap. 4, pp. 61--83. Mucherino, A.; Lavor, C.; Liberti, L.; Maculan, N. (Eds.), Springer, 2013. ISBN 978-1-4614-5127-3, DOI 10.1007/978-1-4614-5128-0_4
A free preprint is available at: https://hal.archives-ouvertes.fr/hal-01955983
