  We show that strong subadditivity provides a simple derivation of the
$g$-theorem for the boundary renormalization group flow in two-dimensional
conformal field theories. We work out its holographic interpretation and also
give a derivation of the $g$-theorem for the case of an interface in
two-dimensional conformal field theories. We also geometrically confirm strong
subadditivity for holographic duals of conformal field theories on manifolds
with boundaries.
