  In this paper we characterise the minimisers of a one-parameter family of
nonlocal and anisotropic energies $I_\alpha$ defined on probability measures in
$\R^n$, with $n\geq 3$. The energy $I_\alpha$ consists of a purely nonlocal
term of convolution type, whose interaction kernel reduces to the Coulomb
potential for $\alpha=0$ and is anisotropic otherwise, and a quadratic
confinement. The two-dimensional case arises in the study of defects in metals
and has been solved by the authors by means of complex-analysis techniques. We
prove that for $\alpha\in (-1, n-2]$, the minimiser of $I_\alpha$ is unique and
is the (normalised) characteristic function of a spheroid. This result is a
paradigmatic example of the role of the anisotropy of the kernel on the shape
of 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 the
interaction potential.
