  We consider Ising models in two and three dimensions, with short range
ferromagnetic and long range, power-law decaying, antiferromagnetic
interactions. We let $J$ be the ratio between the strength of the ferromagnetic
to antiferromagnetic interactions. The competition between these two kinds of
interactions induces the system to form domains of minus spins in a background
of plus spins, or vice versa. If the decay exponent $p$ of the long range
interaction is larger than $d+1$, with $d$ the space dimension, this happens
for all values of $J$ smaller than a critical value $J_c(p)$, beyond which the
ground state is homogeneous. In this paper, we give a characterization of the
infinite volume ground states of the system, for $p>2d$ and $J$ in a left
neighborhood of $J_c(p)$. In particular, we prove that the
quasi-one-dimensional states consisting of infinite stripes ($d=2$) or slabs
($d=3$), all of the same optimal width and orientation, and alternating
magnetization, are infinite volume ground states. Our proof is based on
localization bounds combined with reflection positivity.
