Let the half-sphere in $\mathbb{R}^n$ be defined by $$ R = \{\mathbf{x} \in \mathbb{R}^n : \Vert\mathbf{x}\Vert_2 \leq 1\;\wedge\;x_1 \geq 0\}\,. $$ Any ellipsoid can be defined using a positive definite matrix $\mathbf{A}$ and a center $\mathbf{a}$. The ellipsoid corresponding to $\mathbf{A}$ and $\mathbf{a}$ is $$ E(\mathbf{A,a}) = \{\mathbf{x} \in \mathbb{R}^n : \mathbf{(x-a)}^T\mathbf{A(x-a)} \leq 1\}\,. $$ The question is: for which pair $\mathbf{A}$, $\mathbf{a}$ where $R\subseteq E(\mathbf{A,a})$ is the volume of $E(\mathbf{A,a})$ the smallest? The volume of $E(\mathbf{A,a})$ being given by $$ \text{Vol}(E(\mathbf{A,a})) = \frac{\text{Vol}(S_n)}{\sqrt{\det \mathbf{A}}} $$ where $S_n$ is the $n$-dimensional unit hypersphere.
Looking at the formula for the volume of an ellipsoid it's clear that in order to minimize the volume we have to maximize $\det \mathbf{A}$ subject to $R\subseteq E(\mathbf{A,a})$ for some $\mathbf{a}$. Also it might be helpful that maximizing $\det \mathbf{A}$ is equivalent to maximizing $\log (\det \mathbf{A})$ which, as it turns out, is concave on the domain of positive definite matrices.
So far I've been able to find this exercise on the topic. However,
- I'm not sure if this is the easiest way to answer this question or if it's simply using Slater's condition and the KKT conditions because it wants to practice those.
- The overall structure of the argument is still unclear to me. c) shows that the primal optimum is equal to the dual optimum. And d) shows that the proposed $\overline{\mathbf{A}}$ might be an optimum. How does that imply that $\overline{\mathbf{A}}$ indeed is the optimum?