  We present a metric-space approach to quantify the performance of
density-functional approximations for interacting many-body systems and to
explore the validity of the Hohenberg-Kohn-type theorem on fermionic lattices.
This theorem demonstrates the existence of one-to-one mappings between particle
densities, wave functions and external potentials. We then focus on these
quantities, and quantify how far apart in metric space the approximated and
exact ones are. We apply our method to the one-dimensional Hubbard model for
different types of external potentials, and assess its validity on one of the
most used approximations in density-functional theory, the local density
approximation (LDA). We find that the potential distance may have a very
different behaviour from the density and wave function distances, in some cases
even providing the wrong assessments of the LDA performance trends. We
attribute this to the systems reaching behaviours which are borderline for the
applicability of the one-to-one correspondence between density and external
potential. On the contrary the wave function and density distances behave
similarly and are always sensitive to system variations. Our metric-based
method correctly predicts the regimes where the LDA performs fairly well and
the regimes where it fails. This suggests that our method could be a practical
tool for testing the efficiency of density-functional approximations.
