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.

翻译：本文研究了由配备辛势的Delzant多面体及其对应的环面Kähler流形作为其环面化所产生的对偶平坦空间。我们从环面Kähler几何的视角，在边界上引入了一种对偶平坦结构及与之相关的Bregman散度。我们证明了该散度在边界上的连续性以及推广的毕达哥拉斯定理。此外，我们还给出了环面Kähler流形成为有限集上混合族的环面化的刻画条件。