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A mass at the center of two points (or a 0-sphere) that is gravitationally attracted to both would be in a state of unstable equilibrium; any disturbance would increase the attraction to one and decrease it to the other. Similarly, a mass at the center of a ring (or 1-sphere) would be in a state of unstable equilibrium. However, a mass at the center of a sphere (or 2-sphere) is in a state of neutral equilibrium, because the net force on the mass is zero for all points inside the sphere.

My question is what the state of equilibrium would be for other n-spheres of greater dimensions, such as 3-spheres and 4-spheres.

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  • $\begingroup$ There are two variables here, dimension of the sphere and the exponent in the attraction law. To get neutral equilibrium inside the sphere they need to match, and that's how the $n$-dimensional "gravity" is usually defined to satisfy the Gauss law. $\endgroup$
    – Conifold
    Jun 6, 2017 at 23:47

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