  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), similarly 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 big then the local-global convexity principle breaks down, and the
base spaces can be globally non-convex even for compact manifolds. As one of
surprising examples, 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.
