  Let P be a convex polytope not simple in general. In the focus of this paper
lies a simplicial complex K_P which carries complete information about the
combinatorial type of P. In the case when P is simple, K_P is the same as dP*,
where P* is a polar dual polytope. Using the canonical embedding of a polytope
P into nonnegative orthant, we introduce a moment-angle space Z_P for a
polytope P. It is known, that in the case when P is simple the space Z_P is
homeomorphic to the polyhedral product (D^2,S^1)^{K_P}. When P is not simple,
we prove that the space Z_P is homotopically equivalent to the space
(D^2,S^1)^{K_P}. This allows to introduce bigraded Betti numbers for any convex
polytope. A Stanley-Reisner ring of a polytope P can be defined as a
Stanley-Reisner ring of a simplicial complex K_P. All these considerations lead
to a natural question: which simplicial complexes arise as K_P for some
polytope P? We have proceeded in this direction by introducing a notion of a
polytopic simplicial complex. It has the following property: link of each
simplex in a polytopic complex is either contractible, or retractible to a
subcomplex, homeomorphic to a sphere. The complex K_P is a polytopic simplicial
complex for any polytope P. Links of so called face simplices in a polytopic
complex are polytopic complexes as well. This fact is sufficient enough to
connect face polynomial of a simplicial complex K_P to the face polynomial of a
polytope P, giving a series of inequalities on certain combinatorial
characteristics of P. Two of these inequalities are equalities for each P and
represent Euler-Poincare formula and one of Bayer-Billera relations for flag
f-numbers. In the case when P is simple all inequalities turn out to be
classical Dehn-Sommerville relations.
