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We study the set of ground states $ GS(\\alpha)$ of the potential $\\alpha\\cdot \\Phi$ as a function of the direction vector $\\alpha\\in S^{m-1}$. %We also study the corresponding rotation vectors $\\rv(GS(\\alpha))$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $\\Phi$. 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