# Ellipsoid with minimal volume that contains 2 other ellipsoids

Given two ellipsoids $e_1, e_2$, and we need to find another ellispoid $E$ such that $E=\arg\min_{E\ :\ e_1,e_2\subseteq E} {V(E)}$ (where $V(E)$ is $E$'s volume). In words, $E$ has the minimal volume among all ellipsoids that contain both $e_1$ and $e_2$.

So, I am looking for some algorithm/ formula for computing $E$.

So far, I investigated the case where $e_1$ and $e_2$ have the same center. In that case, I guess that taking the biggest radius (from the union of $e_1$'s radii and $e_2$'s radii), and then projecting the ellipsoids $e_1,e_2$ (or their radii) onto this radius, then repeating this process yields the desired ellipsoid.

However, I have no idea what to do in the general case - where $e_1$ and $e_2$ have different centers.

Any help would be appreciated! Thanks in advance!