  The seminal results of Bourgain, Brezis, Mironescu and D\'avila show that the
classical perimeter can be approximated by a family of nonlocal perimeter
functionals. We consider a corresponding second order expansion for the
nonlocal perimeter functional. In a special case, the considered family of
energies is also relevant for a variational model for thin ferromagnetic films.
We derive the Gamma--limit of these functionals. We also show existence for
minimizers with prescribed volume fraction. For small volume fraction, the
unique, up to translation, minimizer of the limit energy is given by the ball.
The analysis is based on a systematic exploitation of the associated
symmetrized autocorrelation function.
