# Seeking criteria for “threadable” pairs of centrosymmetric polyhedra

Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$: "for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$." Say that $A$ and $B$ are threadable (my terminology) iff there is a scaling and rotation of $B$ to $B'$ such that (a) Every vertex of $A$ is exterior to $B'$, and (b) Every vertex of $B'$ is exterior to $A$. (I am exploring this notion of "threadability" as a measure of shape similarity.)

Two examples. (1) For $A$ a cube (blue) and $B$ a cuboctahedron (red), $(A,B)$ is threadable, e.g.:

Note that $A$ and $B$ are not duals; for duals, threadability is obvious.

(2) For $A$ a truncated icosahedron (red) and $B$ a vertically stretched pentagonal bipyramid (blue), I believe (but have not proved) it is not possible to scale & rotate $B$ to thread with $A$:

Computing whether or not $A$ and $B$ are threadable seems quite difficult, only achievable exactly via an $O(n^k)$ algoithm for $n$-vertex polyhedra, for $k$ an exponent that captures all the combinatorial possibilities. Perhaps $k=6$ would be necessary; I haven't thought that through carefully, but certainly it would a high computational complexity.

So, here, finally, is my question.

Q. Are there succinct sufficiency criteria for when a pair $(A,B)$ are guaranteed to be threadable?

What I have in mind here is something like this: "If the diameter/width ratio of $A$ and $B$ is approximately (or even: exactly) the same, then $A$ and $B$ are threadable." I don't believe this, but it gives the flavor of sufficiency conditions I seek. I have a sense that no such "simple" sufficiency conditions exist, because of the seeming dependence upon the micro- combinatorial structure of $A$ and $B$. But perhaps others can see more clearly through this thicket than I ... ?

It seems to me that two convex polyhedras are threadable if and only if (after scaling and rotation) they can be inscribed in the same strictly convex domain: If $A$ and $B$ are both inscribed in a strictly convex $D$ and two vertices do not meet, then $A$ and $B$ are threaded. If two vertices do coincide, then I guess that by moving $A$ or $B$ a little bit, one should obtain a threading. Conversely, given a threading of $A$ and $B$, one can take the convex hull of the vertices and it should be possible to make it strictly convex by blowing some air in it keeping the vertices fixed.