  Any pure three-qubit state is uniquely characterized by one phase and four
positive parameters. The geometric measure of entanglement as a function of
state parameters can have different expressions. Each of expressions has its
own applicable domain and thus the whole state parameter space is divided into
subspaces that are ranges of definition for corresponding expressions. The
purpose of this paper is to examine the applicable domains for the most general
qubit-interchange symmetric three-qubit states. First, we compute the
eigenvalues of the non-linear eigenvalue equations and the nearest separable
states for the permutation invariant three-qubit states with a fixed phase.
Next, we compute the geometric entanglement measure, deduce the boundaries of
all subspaces, and find allocations of highly and slightly entangled states. It
is shown that there are three applicable domains when the phase factor is
$\pi/2$ while other cases have only two domains. The emergence of the three
domains is due to the appearance of the additional W-state. We show that most
of highly entangled states reside near the boundaries of the domains and states
located far from the boundaries become less-entangled and eventually go to the
product states. The neighbors of W-state are generally more entangled than the
neighbors of Greenberger-Horne-Zeilinger(GHZ) state from the aspect of the
geometric measure. However, the range of the GHZ-neighbors is much more wider
than the range of the W-neighbors.
