  We review the equivalence of maximal center gauge fixing to the problem of
finding the best fit, to a given lattice gauge field, by a thin vortex
configuration. This fit is necessarily worst at the location of P-plaquettes.
We then compare the fits achieved in Gribov copies generated by (i)
over-relaxation; (ii) over-relaxation after Landau gauge preconditioning; and
(iii) simulated annealing. Simulated annealing yields the best fit if all links
on the lattice are included, but the situation changes if we consider only the
lattice volume exterior to P-plaquettes. In this exterior region, the fit is
best for Gribov copies generated by over-relaxation, and worst for Gribov
copies generated after Landau gauge preconditioning. The two fitting criteria
(including or not including the P-plaquettes) yield string tensions differing
by -34% to +20% respectively, relative to the full string tension. Our usual
procedure (``quenched minimization'') seems to be a compromise between these
criteria, and yields string tensions at an intermediate value close to the full
string tension.
