TITLE: What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?
QUESTION [5 upvotes]: The question is mostly in the title:

What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?  

Imagine I have some non-convex polyhedron $P$, and I would like to find the smallest diameter ring that it can pass through in 3-space, undergoing any necessary rotations as it does so.  Is there an efficient way to calculate $D_{ring}$?  Pressing my luck, can I find the set of rotations for $P$ as it passes through the ring?

REPLY [5 votes]: Just a side remark on convex polyhedra: Each of the regular polyhedra except the cube can pass
through a circle of radius smaller than the smallest-radius cylinder circumscribing the polyhedron.
This is proved in Tudor Zamfirescu's delightful paper, "Convex polytopes passing through circles"
(PDF link). There is quite a nice (non-algorithmic) literature on this problem.