The question is mostly in the title:



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



What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?
This is the "piano movers problem", also known as the motion planning problem, which has an enormous literature. Check out
<http://en.wikipedia.org/wiki/Motion_planning>


