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We consider a continuous dynamical system 
$f:X\rightarrow X$ on a compact metric space 
$X$ equipped with an 
$m$-dimensional continuous potential 
$\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{m}):X\rightarrow \mathbb{R}^{m}$. We study the set of ground states 
$GS(\unicode[STIX]{x1D6FC})$ of the potential 
$\unicode[STIX]{x1D6FC}\cdot \unicode[STIX]{x1D6F7}$ as a function of the direction vector 
$\unicode[STIX]{x1D6FC}\in S^{m-1}$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of 
$\unicode[STIX]{x1D6F7}$. In particular, for each 
$\unicode[STIX]{x1D6FC}$ the set of rotation vectors of 
$GS(\unicode[STIX]{x1D6FC})$ forms a non-empty, compact and connected subset of a face 
$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ of the rotation set associated with 
$\unicode[STIX]{x1D6FC}$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in 
$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any 
$m\in \mathbb{N}$ examples with an exposed boundary point (that is, 
$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face 
$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of 
$GS(\unicode[STIX]{x1D6FC})$ is a non-trivial line segment.
