I have an $n$-dimensional ellipsoid $E$ and a hyperplane $H$. This hyperplane cuts $E$ into two parts: $E_1$ and $E_2$ (whose disjoint union is $E$). I want to find another ellipsoid $E'$ that has minimal hyper-volume and contains $E_1$. Is there an efficient algorithm to do this?
My first thought was to formulate it as an optimization problem, but I am having difficulty with formulating it, as I don't know how to formulate the containment ($E_1 \subseteq E'$) constraint.
An approximation for the minimal hyper-volume ellipsoid is also good for my needs.