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Barycenters in the Hellinger–Kantorovich Space

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Abstract

Recently, Liero, Mielke and Savaré introduced the Hellinger–Kantorovich distance on the space of nonnegative Radon measures of a metric space X. We prove that Hellinger–Kantorovich barycenters always exist for a class of metric spaces containing of compact spaces and Polish \(\mathrm{CAT}(1)\) spaces; and if we assume further some conditions on the data, such barycenters are unique. We also introduce homogeneous multimarginal problems and illustrate some relations between their solutions and Hellinger–Kantorovich barycenters. Our results are analogous to the work of Agueh and Carlier for Wasserstein barycenters.

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Acknowledgements

Part of this paper was carried out when N. P. Chung visited University of Science, Vietnam National University at Ho Chi Minh city in summer 2019. He thanks the math department there for its warm hospitality. The authors were partially supported by the National Research Foundation of Korea (NRF) Grants funded by the Korea Government No. NRF-2016R1A5A1008055 , No. NRF-2016R1D1A1B03931922 and No. NRF-2019R1C1C1007107. We thank the anonymous referee for her/his useful comments which vastly improve the paper.

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  1. Department of Mathematics, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon-si, Gyeonggi-do, 16419, Korea

    Nhan-Phu Chung & Minh-Nhat Phung

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Chung, NP., Phung, MN. Barycenters in the Hellinger–Kantorovich Space. Appl Math Optim 84, 1791–1820 (2021). https://doi.org/10.1007/s00245-020-09695-y

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