  We investigate a density-functional theory (DFT) approach for an unpolarized
trapped dilute Fermi gas in the unitary limit . A reformulation of the recent
work of T. Papenbrock [Phys. Rev. A, {\bf 72}, 041602(R) (2005)] in the
language of fractional exclusion statistics allows us to obtain an estimate of
the universal factor, $\xi_{3D}$, in three dimensions (3D), in addition to
providing a systematic treatment of finite-$N$ corrections. We show that in 3D,
finite-$N$ corrections lead to unphysical values for $\xi_{3D}$, thereby
suggesting that a simple DFT applied to a small number of particles may not be
suitable in 3D. We then perform an analogous calculation for the
two-dimensional (2D) system in the infinite-scattering length regime, and
obtain a value of $\xi_{2D}=1$. Owing to the unique properties of the
Thomas-Fermi energy density-functional in 2D our result, in contrast to 3D, is
{\em exact} and therefore requires no finite-$N$ corrections.
