Abstract
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.
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Acknowledgements
The author is indebted to Koichi Tojo for introducing him to the reference [9] and providing valuable comments. He is also grateful to Mao Nakamura and Naomichi Nakajima for fruitful discussions on dually flat spaces. He has gratitude to Mathieu Molitor for giving useful comments on the draft version of this paper. The author is partly supported by Grant-in-Aid for Scientific Research (C) 18K03288.
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Communicated by Hiroshi Matsuzoe.
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Fujita, H. The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry. Info. Geo. (2023). https://doi.org/10.1007/s41884-023-00123-y
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DOI: https://doi.org/10.1007/s41884-023-00123-y