This is a work in progress with Christophe Wacheux, a student of San.
Christophe went to see me in Toulouse to discuss about his thesis. After several discussions, I gave him 2 problems to work on. The first is about semi-local classification of integrable Hamiltonian systems up to exact isomorphisms (i.e. diffeomorphisms which preserve not only the torus fibration and the symplectic form, but also the Hamiltonian vector field). And the second is this one about intrinsic convexity.
Main Theorem: The base space of any almost-toric IHS on a compact connected symplectic manifold is intrinsically convex.
Almost-toric IHS = integrable Hamiltonian system whose singularities are nondegenerate and without hyperbolic components (only elliptic and focus-focus components are allowed)
Recall that the base space of an IHS admits a singular flat integer affine structure.
Intrinsically convex = any 2 points can be connected by an affine geodesic.
A-priori, this work will be part of Christophe’s thesis. I’m giving him some ideas, but he will have to do most of the work :-)