  Under successive Renormalization Group transformations applied to a quantum
state $\ket{\Psi}$ of finite correlation length $\xi$, there is typically a
loss of entanglement after each iteration. How good it is then to replace
$\ket{\Psi}$ by a product state at every step of the process? In this paper we
give a quantitative answer to this question by providing first analytical and
general proofs that, for translationally invariant quantum systems in one
spatial dimension, the global geometric entanglement per region of size $L \gg
\xi$ diverges with the correlation length as $(c/12) \log{(\xi/\epsilon)}$
close to a quantum critical point with central charge $c$, where $\epsilon$ is
a cut-off at short distances. Moreover, the situation at criticality is also
discussed and an upper bound on the critical global geometric entanglement is
provided in terms of a logarithmic function of $L$.
