A divisible convex set O is a convex, bounded, and open subset of an affine chart of the real projective space, on which acts cocompactly a discrete group G of projective transformations. The quotient M=O/G is a closed manifold with a real projective structure. These objects have a rich theory, which involves ideas from dynamical systems, geometric group theory and (G,X)-structures. Moreover, they are an important source of examples of discrete subgroups of Lie groups, and have links with Anosov representations. In dimension 3, there is a strong link between the projective geometry of M and its JSJ and geometric decompositions: M is cut by projectively totally geodesic embedded disjoint tori into pieces admitting a finite-volume hyperbolic structure.
In this talk, we will survey known examples of divisible convex sets, and then describe new examples obtained in collaboration with Gabriele Viaggi, of irreducible, non-symmetric, and non-strictly convex divisible convex sets in arbitrary dimensions (at least 3). Our examples are also new in dimension 3.