Migrational Stability of Plane Tilings

This paper studies a special case of the facility location problem applied to the standard Euclidean plane \(\mathbb {R}^2\) . The agents are uniformly distributed over the plane. The facility for each group is located at its geometric median in order to minimize the average distance between an agent and the facility. We study the equilibrium setting when no agent wishes to change their designated group in order to reduce their expenses (no one wants to selfishly migrate). We provide several useful tools and some results as to what a stable plane tessellation may look like. In particular, we describe all stable tessellations when all groups are triangles.