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In this paper we characterise the minimisers of a one-parameter family ofnonlocal and anisotropic energies $I_\alpha$ defined on probability measures in$\R^n$, with $n\geq 3$. The energy $I_\alpha$ consists of a purely nonlocalterm of convolution type, whose interaction kernel reduces to the Coulombpotential for $\alpha=0$ and is anisotropic otherwise, and a quadraticconfinement. The two-dimensional case arises in the study of defects in metalsand has been solved by the authors by means of complex-analysis techniques. Weprove that for $\alpha\in (-1, n-2]$, the minimiser of $I_\alpha$ is unique andis the (normalised) characteristic function of a spheroid. This result is aparadigmatic example of the role of the anisotropy of the kernel on the shapeof minimisers. In particular, the phenomenon of loss of dimensionality,observed in dimension $n=2$, does not occur in higher dimension at the value$\alpha=n-2$ corresponding to the sign change of the Fourier transform of theinteraction potential.

The equilibrium measure for an anisotropic nonlocal energy