Let $S$ be a rigid body in $\mathbf{R}^3$ of finite diameter. Assume that there is a plane which divides the space in two regions, one containing $S$ in its entirety.
The question is: What is the smallest radius of the circular hole to be in the plane so that $S$ can move in the other region?
Examples: If $S$ is a sphere, it is obviously the radius of the sphere. If $S$ is an ellipsoid with two axes (= rugby ball) is clearly the semi-minor axis. If $S$ is an ellipsoid with 3 axes $a> b> c$ the wanted radius is $b/2$ ... For a right cylinder, it is the generating circle radius. For an oblique cylinder, I do not know.
Second question: how is called this problem in the literature (if any ...)?
Third question: How to calculate the radius for any $S$?
Correction: for the right cylinder of circular section, it is the radius of the disk generator, regardless of the length of the cylinder.
For a helical spring of circular cross section of radius r and major radius R without ties, it is obviously r (since R> r).
What does not work: it is not in general the radius of the circle circumscribing the minimum section (non-convex solid) of radius r or the radius of the circle circumscribing of the largest section.
I have a partial answer to my third question:
x is a point in S and m(x) is the sup of the radius of sphere of center x included in S.
Then the smallest radius of the circular hole is $\ge \sup_{x \in S} m(x)$.
(counter-example for = is the cylinder with radius > height)
Another inequalities is that the smallest radius of the circular hole is greater or equal to the biggest radius of cylinders included in S.
but there is a counter-example to equality : a cylinder with radius > height and small blisters.
