  We report here the first systematic study of the supercluster-void network in
the LCDM concordance cosmology in which voids and superclusters are treated on
an equal footing. Superclusters are defined as individual members of an
over-dense excursion set and voids are defined as individual members of a
complementary under-dense excursion set at the same density threshold. We
determine the geometric, topological and morphological properties of the cosmic
web at a large set of density levels by computing Minkowski functionals for
every supercluster and void using SURFGEN. The properties of the largest
(percolating) supercluster and the complementary void are found to be very
different from properties of individual superclusters and voids. Individual
superclusters totally occupy no more than about 5% of the total volume and
contain no more than 20% of mass if the largest supercluster is excluded.
Likewise, individual voids totally occupy no more than 14% of volume and
contain no more than 4% of mass if the largest void is excluded. Although
superclusters are more massive and voids are more voluminous the difference in
maximum volumes is not greater than by an order of magnitude. The genus value
of individual superclusters can be 5 while the genus of individual voids can
reach 40, implying significant amount of substructure in superclusters and
especially in voids. One of our main results is that large voids, as defined
through the density field (read dark matter distribution) can be distinctly
non-spherical.
