Spatial equilibrium on the plane and an arbitrary population distribution

The existence of immigration proof partition for communities (countries) in a multidimensional space is studied. This is a Tiebout type equilibrium which existence previously was studied under weaker assumptions (measurable density, fixed centers and so on). The migration stability suggests that the inhabitants of frontier have no incentives to change jurisdiction (an inhabitant at every frontier point has equal costs for all possible adjoining jurisdictions). It is required that inter-country border is represented by a continuous curve (surface). Assuming population is distributed in one or two dimension area (convex compact) and this distribution is described by Radon's measure, we prove that for an arbitrary number of countries there exists stable partition into countries. The proof is based on Kakutani's fixed point theorem applied for specific approximation of initial problem with the subsequent passing to the limits.