  We calculate the density profiles of a trapped spin-imbalanced Fermi gas with
attractive interactions in a one-dimensional optical lattice, using both the
local density approximation (LDA) and density matrix renormalization group
(DMRG) simulations. Based on the exact equation of state obtained by Bethe
ansatz, LDA predicts that the gas phase-separates into shells with a partially
polarized core and fully paired wings, where the latter occurs below a critical
spin polarization. This behavior is also seen in numerically exact DMRG
calculations at sufficiently large particle numbers. Unlike the continuum case,
we show that the critical polarization is a non monotonic function of the
interaction strength and vanishes in the limit of large interactions.
