Lowner-John Ellipsoid minimum volume surrounding ellipsoid Consider a set of ellipsoid C. Minimum volume ellipsoid \epsilon with C \subset \epsilon. We can parameterize \epsilon as \epsilon = \left\{v \mid \norm{Av + b}_{2} \leq 1\right\}; where we assume A \in \mathcal{S}_{++}^{n}. The volume is proportional to \text{det} A^{-1}. Thus to find minimal-volume ellipsoid, solve:
\begin{align} \min_{A,b}\quad & \log \text{det} A^{-1} \\ \textrm{s.t.} \quad & \text{sup}_{v \in C} \norm{A v + b}_{2} \leq 1 \end{align}
OR, for finite sets:
\begin{align} \min_{A,b}\quad & \log \text{det} A^{-1} \\ \textrm{s.t.} \quad & \norm{A x_{i} + b }_{2} \leq 1, i = 1 \dots m \end{align}
Inside maximum volume inscribing ellipsoid Consider a set of ellipsoid C. Minimum volume ellipsoid \epsilon with C \subset \epsilon. We can parameterize \epsilon as \epsilon = \left\{Bu + d \mid \norm{u}_{2} \leq 1\right\}; where we assume B \in \mathcal{S}_{++}^{n}.
\begin{align} \max_{B,d}\quad & \log \text{det} B \\ \textrm{s.t.} \quad & \text{sup}_{\norm{u}_{2}} \leq 1, I_{C}\left(Bu+d\right) \leq 0 \end{align}
where I_{C} = 0 when c \in C, \infty otherwise.