Scaling properties of centering forces

Motivated by the centering of biological objects in large cells, we study the generic properties of centering forces inside a ball (or a volume of spherical topology) in $n$ dimensions. We consider two scenarios : autonomous centering (in which distance information is integrated from the agent perspective) and non-autonomous centering (in which distance to the surface is integrated over the whole surface). We find relations between the net centering force and the mean distance$^p$ to the surface. This allows us to find simple scaling laws between the centering force and the distance to the center, as a function of the dimensionality $n$. Interestingly, if the interactions between the agent and the surface are hyper-elastic, the net centering force can still be sub-elastic in the case of autonomous centering. These scaling laws are increasingly violated as the space becomes less convex. Generically, neither scenarios exactly converge to the center of mass of the space.