Find a set of ellipsoids $\{\mathcal E_k\}_{k=1}^N$ such that their union contains a convex set $\mathcal P$, i.e., $$\mathcal P\subseteq\bigcup_{k=1}^N\mathcal E_k$$
I am wondering (because Google is not helping) if there are algorithms out there that can solve (variations of) this problem. From what I understand, this is an NP-hard problem. However, I do not know what the common name for it is - and am not able to find any papers that tackle algorithms/heuristics to solve the problem. Are there any?
Clarification: assume that there are constraints on the size of $\mathcal E_k$. For example, they are generated by a black box method that generates ellipsoids of contant volume, but random shape (rotation, principal axes).