Why the hoop conjecture is not stated for a sphere? I know about the Kip Thorne's hoop conjecture only in very general terms. What I cannot figure out is why the conjecture is stated for a rotated hoop instead of a sphere, since in my naive perception they are almost the same.
 A: You are quite correct that the hoop conjecture is basically the sphere conjecture. That is, the conjecture states that for a black hole to be formed all the matter must fit inside the boundary formed when the hoop is spun i.e. a sphere.
Why Thorne chose to name the conjecture as he did only he knows. Maybe he just liked the name better. However one possible reason occurs to me:
When working with static black holes we often use the Schwarzschild coordinates $(t,r,\theta,\phi)$. The $r$ coordinate looks like a radius, but is not. More precisely it is not the distance you would measure if you lowered a tape measure down towards the black hole. Instead if you measure the circumference, $C$, of a circle centred on the black hole then the Schwarzschild $r$ parameter is defined as:
$$ r = \frac{C}{2\pi} $$
In flat spacetime $r$ is indeed just the distance from the edge of the circle to its centre, but in curved spacetime $r$ can be greater or less than this distance.
The point of the hoop is that this is a circumference, and is therefore automatically related to the Schwarzschild radius as above. This is true of a sphere as well of course, but there are many circles you can draw on a sphere and only a subset of them are circumferences. With a hoop the circumference is unambiguous.
