Convexity of singular affine structures and toric focus integrable Hamiltonian systems This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners) analogous to the toric case but their associated integral affine structures are singular with non trivial monodromy due to focus singularities. We obtain a series of convexity results both positive and negative for such singular integral affine base spaces. In particular near a focus singular point they are locally convex and the local global convexity principle still applies. They are also globally convex under some natural additional conditions. However when the monodromy is sufficiently large the local global convexity principle breaks down and the base spaces can be globally non convex even for compact manifolds. As a surprising example we construct a 2 dimensional integral affine black hole which is locally convex but for which a straight ray from the center can never escape