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arXiv · 2305.08422

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

Hajime Fujita
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Abstract

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.

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Hajime Fujita. 2023-05-15. The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry. https://doi.org/10.1007/s41884-023-00123-y

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