when does moving the segment increase the potential, and when does moving the segment decrease the potential?
That's a good approach. If the segment is closer to the point of interest after it is moved, the potential well is deeper. Which tells you that it is deepest at the center.
I sit corrected; my argument that as you enter the cylinder you can ignore the outer parts breaks down.
As mister shminux mentioned somewhere, he is happy and qualified to answer questions in the field of the Relativity. Here is mine:
A long rod (a cylinder) could have a large escape velocity in the direction of its main axe. From its end, to the "infinity". Larger than the speed of light. While the perpendicular escape velocity is lesser than the speed of light.
Is this rod then an asymmetric black hole?