Ground states are accumulation points of equilibrium states (associated with a certain potential) when the temperature goes to zero. While they have been studied in the physics literature for quite some time, their rigorous mathematical treatment has just begun in the last 15 years or so. Ground states are also particular examples of maximizing measures that are studied in ergodic optimization. In this talk we consider multi-dimensional potentials and address the problem of maximizing the coordinate integrals simultaneously. We show that this problem is naturally related to the geometry of the associated rotation sets.