**Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through?**


*i.e. which are the necessary and sufficient conditions for a planar compact set C to pass through a closed interval in a plane?*


The matter has been studied in the 1982 paper by Gilbert Strang, "[The width of a chair](https://www.jstor.org/stable/2320823)," in Amer. Math. Monthly; but only ends in certain (interesting) *conjectures* in case of a non convex C (the problem in case of a convex C does get a definitive answer as its shortest orthogonal chord).


A development over this question would be the still open [moving sofa problem](https://en.wikipedia.org/wiki/Moving_sofa_problem) (Leo Moser's 1966 problem, "[Moving Furniture through a Hallway](https://www.jstor.org/stable/2028150)," Problem 66-11 in SIAM Review).



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> Does the question have a definitive answer? Any references/latest work done with regards to the question are welcome.
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Passing C through a slot
Concerning nonconvex $C$, I posed it as an open-problem exercise in *[Computational Geometry in C](http://cs.smith.edu/%7Ejorourke/books/compgeom.html)* to determine the worst polygon (or, equivalently, the worst polygonal
curve $C$), worst in the number of "moves," to get $C$ through the doorway interval
(p.321, Ex.4). I defined a "move" as monotonic $x$ and $y$ translation and $\theta$ rotation. The idea was to try to find a shape that requires many reversals.
At that time the number of "backups" was only known to lie between $\Omega(n)$ and $O(n^2)$ moves
for a polygon of $n$ vertices, and I am unaware of any subsequent improvements.


For the $O(n^2)$ upper bound, see:



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> Yap, Chee-Keng. "How to move a chair through a door." *IEEE Journal of Robotics and Automation* (1987): 172-181. ([IEEE link](https://ieeexplore.ieee.org/document/1087093))
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This paper also computes the minimum door width for an $n$-gon with an $O(n^2)$-time algorithm.


