  We address the issue of point value reconstructions from cell averages in the
context of third order finite volume schemes, focusing in particular on the
cells close to the boundaries of the domain. In fact, most techniques known in
the literature rely on the creation of ghost cells outside the boundary and on
some form of extrapolation from the inside that, taking into account the
boundary conditions, fills the ghost cells with appropriate values, so that a
standard reconstruction can be applied also in boundary cells. In (Naumann,
Kolb, Semplice, 2018), motivated by the difficulty of choosing appropriate
boundary conditions at the internal nodes of a network, a different technique
was explored that avoids the use of ghost cells, but instead employs for the
boundary cells a different stencil, biased towards the interior of the domain.
  In this paper, extending that approach, which does not make use of ghost
cells, we propose a more accurate reconstruction for the one-dimensional case
and a two-dimensional one for Cartesian grids. In several numerical tests we
compare the novel reconstruction with the standard approach using ghost cells.
