Torification of Delzant polytope as dually flat space and its applications

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