What is the shape of the intersection of all projections of a cube in $\mathbb{R}^3$?

Question

Let $Q\subset \mathbb{R}^3$ denote the cube $[-1,1]\times[-1,1]\times[-1,1]$.

Let $\xi\in\mathbb{S}^2$ be a vector on the unit sphere centered at the origin.

Let $\xi^{\perp}$ denote the linear subspace that is orthogonal to the vector $\xi$.

Denote $Q\mid_{\xi^{\perp}}$ as the projection of $Q$ onto the plane $\xi^{\perp}$.

Question: What shape is $\bigcap_{\xi\in\mathbb{S}^2}Q\mid_{\xi^{\perp}}$?

Answer 1

I don't think you've "mathematized" this correctly. Without even bothering with projecting $Q$, isn't the intersection of all the $\xi^{\perp}$ just the origin?

So you probably want to consider all $SO(3)$ rotations of the cube, and project them all onto a fixed plane, say the $x−y$ plane.

Once you've fixed this issue, the resulting intersection will be rotation invariant, as any element of $SO(3)$ can be composed with a suitable rotation around the $z$-axis. Additionally, rotating and projecting a convex set yields a convex set, and intersecting a bunch of convex sets yields a convex set. That doesn't seem to leave a lot of possibilities (hint, hint).

If you already knew all that, and were just curious about the resulting radius, take a look at the largest sphere you can inscribe in your cube.

Answer 2

Another possibility (which may or not be what the OP meant) …

For each given $\xi$, let $E_\xi$ be the Minkowski sum of $Q$ and the infinite line $L_\xi$ through the origin in the direction of $\xi$. Informally, $E_\xi$ is the shape that you get by sweeping $Q$ along $L_\xi$. It’s what CAD folks would call an extruded shape.

Then the intersection of all the $E_\xi$ is something interesting. Clearly it contains the unit sphere (since this is a subset of each $E_\xi$). I’d guess that the intersection is actually equal to the unit sphere. Given a point outside the unit sphere, it seems fairly clear that you can construct an $E_\xi$ that doesn’t contain it.

Something more general might be true: the intersection obtained from any convex body (not just a cube) might be the largest sphere contained within that body.

Edit:
Based on the comment below, the smallest enclosing sphere seems more likely. Willing to put more thought into this if the OP confirms that I’m solving the right problem.