Given 0 < s < d/2 with s ≤ 1, we are interested in the large N-behavior of the optimal constant κN in the Hardy inequality ΣNn=1(-Δn)s ≥ κN Σn<m |Xn - Xm|-2s, when restricted to antisymmetric functions. We show that N1-2s/d κN 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.