The Width of a Chair

TLDR: For a convex set intuition suggests that a reversal of this kind, and even a rotation, should not be necessary: if the set can get through at all, it can do so without turning.

Abstract: Moving a large chair is not much fun. That is especially true when a door is involved. Probably the designer verifies that his chair will go through a standard door, but he never supplies instructions, and trial and error is the algorithm most frequently adopted. At present we do not know a uniformly successful alternative, but at least we can call attention to the problem and give a partial solution. In the case of a convex chair, the problem is easier but still not completely solved. We begin in two dimensions, with a compact set C and the closed interval I = {(x, y): x = 0, 0 < y < 1}. The goal is to determine necessary and sufficient conditions for C to pass through I by a continuous family R, of rigid motions-translations combined with rotations. It is like mailing a postcard of shape C into a slot I, and the motion can bring points of C back through the slot before the whole set ultimately passes through. For a convex set intuition suggests that a reversal of this kind, and even a rotation, should not be necessary: if the set can get through at all, it can do so without turning. We anticipate a single rotation, to make C as thin as possible in the vertical direction, and a single translation to put C to the left of I. Then if C can pass through I, translation in the x direction should do it. Our contribution is to confirm that this intuition is correct in the plane. In three dimensions it is not correct. Harold Stark has constructed convex chairs which can pass through a door, either square or circular, although no projection of the chair will fit in the doorway. The chair itself can go through, but it cannot be put into a box (or a cylinder) that will. I am extremely grateful to the referee who observed that my original example was unsatisfactory, and to Harold Stark and Mike Artin for correcting it. The smallest box into which it will fit is not known.