Consider the following optimization problem:
Input: a 3-dimensional "object" $O$.
Output: a covering of $O$ by a list of $k$ spheres $S_1, \ldots, S_k$ (given by their centers and radii) minimizing the volume of $(\bigcup_i S_i) - O$.
Increasing $k$ results in better optimum (smaller wasted space). I am interest in the trade-off between $k$ and the optimum wasted space. Has this problem been studied before?