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

翻译：本文研究由配备辛势的Delzant多面体及其对应的环面Kähler流形（作为其环面化）所导出的对偶平坦空间。我们从环面Kähler几何的角度，在边界上引入对偶平坦结构及其关联的Bregman散度。我们证明了边界上散度的连续性及广义勾股定理。同时，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。