  In the paper we treat Gale diagrams in a combinatorial way. The
interpretation allows to describe simplicial complexes which are Alexander dual
to boundaries of simplicial polytopes and, more generally, to nerve-complexes
of general polytopes. This technique and recent results of N.Yu.Erokhovets are
combined to prove the following: Buchstaber invariant $s(P)$ of a convex
polytope equals 1 if and only if $P$ is a pyramid. In general, we describe a
procedure to construct polytopes with $s_R(P)>k$. The construction has purely
combinatorial consequences. We also apply Gale duality to the study of bigraded
Betti numbers and f-vectors of polytopes.
