Supposed you to want find out if an ellipsoid fully contains a convex shape. In the picture below I have drawn two situations. The left shape is overlapping
with the ellipsoid, and the right shape is fully contained in it.
Using an algorithm like the GJK Expanding Polytope algorith (see e.g. 'Collision Detection in Interactive 3D Environments'), we can get a penetration depth vector, $\vec{d}$. Using the support function of the shape ($S$), and the ellipsoid ($E$), we can find the extreme points of the shape along the penetration depth vector. If S is fully contained, then $S_{max} < E_{max}$ and $S_{min} > E_{min}$, as is seen in the right-hand of the picture (in red), and where $S_{max} = s_S(\vec{d})$, $S_{min} = s_S(-\vec{d})$, and analogously for shape $E$.
I have drawn a 2D example for simplicity, but I'm mostly interested in the 3D case. I can't find anything wrong with this approach, but it would be quite slow for my needs. Unfortunately, I don't know the formal name of this kind of problem and could thus not find any other solutions that might have a better performance. It would be nice if any alternative approaches would work for general convex-shaped containers.