Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I want to compute the smallest volume ellipsoid $E$ containing $P$ (that is, $P \subseteq E$) in polynomial time.
This problem has been well studied for the case where polytope $P$ is given as the convex hull of a set of points (see this and this reference, for example). However, in my case $P$ is described as an intersection of half-planes, so I cannot use the previous references unless I compute all the vertices of $P$, which has exponential complexity and, thus, is not an option.
Are there any references or research for this specific instance of the problem, where polytope $P$ is described as an intersection of half-planes?
