  Inspired by recent simulations on highly open liquid crystalline structures
formed by rigid planar nanorings we present a simple theoretical framework
explaining the prevalence of smectic over nematic ordering in systems of
ring-shaped objects. The key part of our study is a calculation of the excluded
volume of such non-convex particles in the limit of vanishing thickness to
diameter ratio. Using a simple stability analysis we then show that dilute
systems of ring-shaped particles have a strong propensity to order into smectic
structures with an unusual antinematic order while solid disks of the same
dimensions exhibit nematic order. Since our model rings have zero internal
volume these smectic structures are essential empty, resembling the strongly
porous structures found in simulation. We argue that the antinematic
intralamellar order of the rings plays an essential role in stabilizing these
novel smectic structures.
