  We discuss the local density approximation approach to calculating the ground
state energy of a one-dimensional Fermi gas containing a single impurity, and
compare the results with exact numerical values that we have for up to 11
particles for general interaction strengths and up to 30 particles in the
strongly interacting case. We also calculate the contact coefficient in the
strongly interacting regime. The different theoretical predictions are compared
to recent experimental results with few-atom systems. Firstly, we find that the
local density approximation suffers from great ambiguity in the few-atom
regime, yet it works surprisingly well for some models. Secondly, we find that
the strong interaction theories quickly break down when the number of particles
increase or the interaction strength decreases.
