  The corner polyhedron is described by minimal valid inequalities from maximal
lattice-free convex sets. For the Relaxed Corner Polyhedron (RCP) with two free
integer variables and any number of non-negative continuous variables, it is
known that such facet defining inequalities arise from maximal lattice-free
splits, triangles and quadrilaterals. We improve on the tightest known upper
bound for the approximation of the RCP, purely by minimal valid inequalities
from maximal lattice-free quadrilaterals, from 2 to 1.71. We generalize the
tightest known lower bound of 1.125 for the approximation of the RCP, purely by
minimal valid inequalities from maximal lattice-free triangles, to an infinite
subclass of quadrilaterals.
