  Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider
a real projective manifold M which is obtained by gluing together the polytopes
in P along their facets in such a way that the union of any two adjacent
polytopes sharing a common facet is convex. We prove that the real projective
structure on M is (1) convex if P contains no triangular polytope, and (2)
properly convex if, in addition, P contains a polytope whose dual polytope is
thick. Triangular polytopes and polytopes with thick duals are defined as
analogues of triangles and polygons with at least five edges, respectively.
