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I'm trying to study Monge-Kantorovich formulation for optimal transport. But I'm interested more in knowing the metrics that can be defined using such formulation, also interested in duality and in some applications.

So, I would like to know some references where I can study the formulation but more specifically the above topics.

Thank you in advance.

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  • $\begingroup$ Nearly all books on OT will cover the Wasserstein metric, the dual problem, and applications... I suggest Santambrogios book, or maybe Figalli's new book `introduction to optimal transport'. $\endgroup$ Commented Sep 1, 2022 at 9:17

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Computational Optimal Transport, by Gabriel Peyré and Marco Cuturi covers many topics, including Metric Properties of Optimal Transport, as well as Duality. It also covers "applications", if you consider rather theoretical applications to be applications.

Be warned that it is fairly advanced mathematically.

Authors' summary:

This book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications. Our focus is on the recent wave of efficient algorithms that have helped translate attractive theoretical properties onto elegant and scalable tools for a wide variety of applications. We also give a prominent place to the many generalizations of OT that have been proposed in but a few years, and connect them with related approaches originating from statistical inference, kernel methods and information theory.

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