Let $Q$ be a cube in $\mathbb{R}^3$ centered at the origin with side length $l$.
Let $\psi\in\text{SO}(3)$ so that $\psi(Q)$ is a rotation of $Q$ in the space $\mathbb{R}^3$.
Denote $\pi(\psi(Q))$ as the projection of the rotated cube onto the $xy-$coordinate plane. By projection, I mean I am looking at the shadow that $\psi(Q)$ casts perpendicular to the plane.
Question: What is the radius of the largest disc inscribed in the projection $\pi(\psi(Q))$?
Follow up question: Denote this inscribed disc as $D_l$. What shape is the pre-image $\pi^{-1}(D_l)$ inside of $\psi(Q)$? Is it always an ellipsoid?