  Given $0<s<\frac d2$ with $s\leq 1$, we are interested in the large
$N$-behavior of the optimal constant $\kappa_N$ in the Hardy inequality
$\sum_{n=1}^N (-\Delta_n)^s \geq \kappa_N \sum_{n<m} |X_n-X_m|^{-2s}$, when
restricted to antisymmetric functions. We show that $N^{1-\frac{2s}d}\kappa_N$
has a positive, finite limit given by a certain variational problem, thereby
generalizing a result of Lieb and Yau related to the Chandrasekhar theory of
gravitational collapse.
