A convexity theorem for real projective structures

Abstract

Given a finite collection P of convex n-polytopes in RP n (n ≥ 2), 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.

Around groups in Hilbert geometry

“…Nevertheless, thanks to Lemma 4.4, Lee shows: Theorem 4.5 (Lee [55] or [56]). Let Γ be a discrete group of SL d+1 (R) acting on a properly convex open set Ω.…”

“…Indeed, in the case of Hilbert geometry the two connected component given by a bisector have no reason to be convex. Nevertheless, thanks to Lemma 4.4, Lee shows: Theorem 4.5 (Lee [55] or [56]). Let Γ be a discrete group of SL d+1 (R) acting on a properly convex open set Ω.…”