$$\begin{array}{ll} & \underset{w \in \mathbb R^n}{\text{minimize}} & & \quad w^{H} w \\ & \text{subject to} & & w^{H} A w \geq 1 \\ & & & w^{H} B w \geq 1 \end{array} \\$$
where $A$ and $B$ are positive semidefinite. Can I get a closed-form expression for the optimal $w$?