  We study the geometrical entanglement of the XXZ chain in its critical
regime. Recent numerical simulations [Q.-Q. Shi, R. Or\'us, J. O. Fj\ae restad
and H.-Q Zhou, New J. Phys. {\bf 12}, 025008 (2010)] indicate that it scales
linearly with system size, and that the first subleading correction is
constant, which was argued to be possibly universal. In this work, we confirm
the universality of this number, by relating it to the Affleck-Ludwig boundary
entropy corresponding to a Neumann boundary condition for a free compactified
field. We find that the subleading constant is a simple function of the
compactification radius, in agreement with the numerics. As a further check, we
compute it exactly on the lattice at the XX point. We also discuss the case of
the Ising chain in transverse field and show that the geometrical entanglement
is related to the Affleck-Ludwig boundary entropy associated to a ferromagnetic
boundary condition.
