  In this paper the geometric entanglement (GE) of systems in one spatial
dimension (1D) and in the thermodynamic limit is analyzed focusing on two
aspects. First, we reexamine the calculation of the GE for
translation-invariant matrix product states (MPSs) in the limit of infinite
system size. We obtain a lower bound to the GE which collapses to an equality
under certain sufficient conditions that are fulfilled by many physical
systems, such as those having unbroken space (P) or space-time (PT) inversion
symmetry. Our analysis justifies the validity of several derivations carried
out in previous works. Second, we derive scaling laws for the GE per site of
infinite-size 1D systems with correlation length $\xi \gg 1$. In the case of
MPSs, we combine this with the theory of finite-entanglement scaling, allowing
to understand the scaling of the GE per site with the MPS bond dimension at
conformally invariant quantum critical points.
