Check if a $n$-dimensional hypercuboid fits through $n-1$ dimensional hypersphere Starting with 3 dimensions, let's say I have a cuboid with the edge lengths represented as $[a,b,c]$. I want to check if this cuboid can fit through a 2D circle with the diameter of $d$, based on the three primary projections (top, side, front). I can do it by finding all (in this case three) unique pairs of edges from $[a,b,c]$ that represent unique faces of the cuboid and compare the diagonal of each face with the diameter of the circle.
Can I apply the same reasoning when scaling both the cuboid and the "hole" to higher dimensions (while the number of dimensions of the cuboid is $N$ and the number of dimensions of the "hole" is $N-1$, always smaller by 1 as my professor suggested)?
Can I represent the edges of the hypercuboid as $[a,b,c,...]$ for $N$ dimensions, find the unique $N-1$-dimensional projections using the combinations of the edges and compare the diagonals of these projections with the diameter of a $N-1$-dimensional hypersphere?
 A: This may be more of a too-long-for-a-comment than a complete answer; still hoping to leave you some fun in working out details to your satisfaction.
To fix notation and terminology, let's say we have $n > 1$ positive real numbers $a_{k}$ for $1 \leq k \leq n$. Let's call the cuboid the $n$-fold Cartesian product
$$
C = \prod_{k=1}^{n} [0, a_{k}] = [0, a_{1}] \times [0, a_{2}] \times \cdots \times [0, a_{n}].
$$
Let $m$ be an integer with $1 \leq m < n$. (In the comments I was assuming $m = n - 1$.) Let's define an $m$-dimensional slice of $C$ to be the $m$-dimensional cuboid we obtain by selecting one of $\binom{n}{m}$ sets of $m$ coordinates and replacing the other $n - m$ factors in the product by the singleton set $\{0\}$.
If the lengths of the sides are denoted $b_{1}$, $b_{2}$, \dots, $b_{m}$, the diameter of the slice is
$$
\sqrt{b_{1}^{2} + b_{2}^{2} + \cdots + b_{m}^{2}} = \biggl(\sum_{\ell=1}^{m} b_{\ell}^{2}\biggr)^{1/2}.
$$
Fix a positive radius $r$, and let $W_{m,r}$ be the "wall" consisting of all points $(x_{k})_{k=1}^{n} = (x_{1}, \cdots x_{n})$ in Cartesian $n$-space such that
$$
r^{2} < x_{1}^{2} + x_{2}^{2} + \cdots + x_{m}^{2} = \sum_{k=1}^{m} x_{k}^{2}.
$$
Geometrically, the wall is the result of "boring an $m$-dimensional hole of radius $r$ out of $n$-space. Alternatively, we can "slice" the wall by setting $x_{m+1} = \cdots = x_{n} = 0$, namely, by intersecting with the copy of $m$-space embedded in $n$-space as the set of points where the first $m$ coordinates vary freely and the others are $0$.

The question is, "Under what conditions can we move the cuboid, in an arbitrary axis orientation, by translations through the sliced wall?" Equivalently, "Under what conditions can we place a translated copy of $C$ (after an arbitrary axis-permuting rotation) inside the complement of $W_{m,r}$, i.e., inside the Cartesian product of the $m$-ball and $(n - m)$-dimensional space?"
The necessary and sufficient condition is, every $m$-dimensional slice of $C$ has diameter at most $2r$. In a specific example, it suffices to pick "the $m$ largest lengths."
