  We study a geometric variational problem for sets in the plane in which the
perimeter and a regularized dipolar interaction compete under a mass
constraint. In contrast to previously studied nonlocal isoperimetric problems,
here the nonlocal term asymptotically localizes and contributes to the
perimeter term to leading order. We establish existence of generalized
minimizers for all values of the dipolar strength, mass and regularization
cutoff and give conditions for existence of classical minimizers. For
subcritical dipolar strengths we prove that the limiting functional is a
renormalized perimeter and that for small cutoff lengths all mass-constrained
minimizers are disks. For critical dipolar strength, we identify the next-order
$\Gamma$-limit when sending the cutoff length to zero and prove that with a
slight modification of the dipolar kernel there exist masses for which
classical minimizers are not disks.
