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  On isometries of compact Lp Wasserstein spaces

Santos-Rodríguez, J. (2022). On isometries of compact Lp Wasserstein spaces. Advances in Mathematics, 409(Part A): 108632. doi:10.1016/j.aim.2022.108632.

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Item Permalink: https://hdl.handle.net/21.11116/0000-000B-1D78-F Version Permalink: https://hdl.handle.net/21.11116/0000-000E-2CBA-0
Genre: Journal Article
Latex : On isometries of compact $L^p$–Wasserstein spaces

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https://doi.org/10.1016/j.aim.2022.108632 (Publisher version)
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https://doi.org/10.48550/arXiv.2102.08725 (Preprint)
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 Creators:
Santos-Rodríguez, Jaime1, Author           
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Metric Geometry, Differential Geometry
 Abstract: Let $(X,d,\mathfrak{m})$ be a metric measure space. The study of the
Wasserstein space $(\mathbb{P}_p(X),\mathbb{W}_p)$ associated to $X$ has proved
useful in describing several geometrical properties of $X.$ In this paper we
focus on the study of isometries of $\mathbb{P}_p(X)$ for $p \in (1,\infty)$
under the assumption that there is some characterization of optimal maps
between measures, the so called Good transport behaviour $GTB_p$. Our first
result states that the set of Dirac deltas is invariant under isometries of the
Wasserstein space. Additionally we obtain that the isometry groups of the base
Riemannian manifold $M$ coincides with the one of the Wasserstein space
$\mathbb{P}_p(M)$ under assumptions on the manifold; namely, for $p=2$ that the
sectional curvature is strictly positive and for general $p\in (1,\infty)$ that
$M$ is a Compact Rank One Symmetric Space.

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Language(s): eng - English
 Dates: 2022
 Publication Status: Issued
 Pages: -
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 2102.08725
DOI: 10.1016/j.aim.2022.108632
 Degree: -

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Title: Advances in Mathematics
Source Genre: Journal
 Creator(s):
Affiliations:
Publ. Info: Elsevier
Pages: - Volume / Issue: 409 (Part A) Sequence Number: 108632 Start / End Page: - Identifier: -