The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry

In this paper we study dually flat spaces arising from Delzant polytopes equipped with a symplectic potential together with their corresponding toric Kähler manifolds as their torifications. We introduce a dually flat structure and the associated Bregman divergence on the boundary from the viewpoint of toric Kähler geometry. We show a continuity and a generalized Pythagorean theorem for the divergence on the boundary. We also provide a characterization for a toric Kähler manifold to become a torification of a mixture family on a finite set.