  In this paper we consider shape optimisation problems for sets of prescribed
mass, where the driving energy functional is nonlocal and anisotropic. More
precisely, we deal with the case of attractive/repulsive interactions in two
and three dimensions, where the attraction is quadratic and the repulsion is
given by an anisotropic variant of the Coulomb potential. Under the sole
assumption of strict positivity of the Fourier transform of the interaction
potential, we show the existence of a threshold value for the mass above which
the minimiser is an ellipsoid, and below which the minimiser does not exist.
If, instead, the Fourier transform of the interaction potential is only
nonnegative, we show the emergence of a dichotomy: either there exists a
threshold value for the mass as in the case above, or the minimiser is an
ellipsoid for any positive value of the mass.
