A convexity theorem for real projective structures

Abstract

Given a finite collection $$\mathcal {P}$$ of convex n-polytopes in $$\mathbb {R}\hbox {P}^n$$ ( $$n\ge 2$$ ), we consider a real projective manifold M which is obtained by gluing together the polytopes in $$\mathcal {P}$$ 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 $$\mathcal {P}$$ contains no triangular polytope, and (2) properly convex if, in addition, $$\mathcal {P}$$ 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.