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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?

VShaw
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  • Why would the pre-image be an ellipsoid? It seems to me to be the intersection between a cylinder and $\psi(Q)$, and never an ellipsoid. – Ron Kaminsky Nov 20 '23 at 21:33
  • An obvious lower bound for the radius is half the length of the side, because the projection of π(S) where S is the inscribed sphere in Q is a disc of that radius which is contained in the projection in question. And an obvious upper bound is $\sqrt{3}$ times that lower bound. – Ron Kaminsky Nov 24 '23 at 13:37

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