A convexity theorem for real projective structures

Abstract

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.

61 pages, 19 figures

References in corpus (2)

• 3-manifolds with(out) metrics of nonpositive curvature
• Convex projective structures on Gromov--Thurston manifolds

Cited by in corpus (3)

• The classification of ends of properly convex real projective orbifolds II: Properly convex radial ends and totally geodesic ends
• A classification of radial and totally geodesic ends of properly convex real projective orbifolds
• The convex real projective orbifolds with radial or totally geodesic ends: The closedness and openness of deformations