A convexity theorem for real projective structures

Given a finite collection of convex n-polytopes in (), we consider a real projective manifold M which is obtained by gluing together the polytopes in 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 contains no triangular polytope, and (2) properly convex if, in addition, 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.