  The classical Oseen-Frank theory of liquid crystal elasticity is based on the
experimentally verified fact that there are three independent modes of
distortion, each with its associated elastic constant. On the other hand, the
arguably more first-principles order parameter-based Landau-de Gennes theory
only involves two independent elastic modes. The resulting 'elastic constants
problem' has led to a considerable amount of vexation among theorists. In a
series of papers at the turn of the century, Fukuda and Yokoyama suggested that
the resolution of this problem could be found in the proper treatment of
non-local effects in the ideal part of the free energy. They used an ingenious,
but technically complex, technique based on a field-theoretic approach to
semi-flexible polymers. Here we revisit their idea but now in the more
accessible framework of density functional theory of rigid particles. Our work
recovers their main results for rod-like particles, in that generically an
ordered assembly of non-interacting rods has three independent elastic
constants associated to it that all scale as the square of the length of the
particles and obey the inequalities $K_2 < K_1 < K_3$. We also consider the
case of disc-like particles, and then find in line with expectations that $K_3
< K_1 < K_2$.
