  In this paper we consider a general class of anisotropic energies in three
dimensions and give a complete characterisation of their minimisers. We show
that, depending on the Fourier transform of the interaction potential, the
minimiser is either the normalised characteristic function of an ellipsoid or a
measure supported on a two-dimensional ellipse. In particular, it is always an
ellipsoid if the transform is strictly positive, while when the Fourier
transform is degenerate both cases can occur. Finally, we show an explicit
example where loss of dimensionality of the minimiser does occur.
