We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul, which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an open subset of the representation variety. We also give a relative version for noncompact (G;X) manifolds of the openness of their holonomies.

We describe an algorithm to compute trisections of orientable four-manifolds using arbitrary triangulations as input. This results in explicit complexity bounds for the trisection genus of a 4-manifold in terms of the number of pentachora (4-simplices) in a triangulation.