A note on flatness of non separable tangent cone at a barycenter

Thibaut Le Gouic

doi:10.5802/crmath.66

Géométrie, Probabilité

A note on flatness of non separable tangent cone at a barycenter
[Une note sur la platitude du cône tangeant à un barycentre]

Thibaut Le Gouic ¹

¹ Massachusetts Institute of Technology, Department of Mathematics and Centrale Marseille, I2M, UMR 7373, CNRS, Aix-Marseille univ., Marseille, 13453, France

Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 489-495.

Résumé
Abstract

Étant donné une mesure de probabilité $P$ sur un espace d’Alexandrov $S$ avec courbure minorée, nous prouvons que le support de la mesure poussée de $P$ sur le cône tangent $T_{b^{☆}} S$ à son barycentre (exponentiel) $b^{☆}$ est un sous-ensemble d’un espace de Hilbert, sans condition de séparabilité du cône tangent.

Given a probability measure $P$ on an Alexandrov space $S$ with curvature bounded below, we prove that the support of the pushforward of $P$ on the tangent cone $T_{b^{☆}} S$ at its (exponential) barycenter $b^{☆}$ is a subset of a Hilbert space, without separability of the tangent cone.

Reçu le : 2019-09-16
Révisé le : 2019-12-31
Accepté le : 2020-04-29
Publié le : 2020-07-28

DOI : 10.5802/crmath.66

Affiliations des auteurs :

Thibaut Le Gouic ¹

¹ Massachusetts Institute of Technology, Department of Mathematics and Centrale Marseille, I2M, UMR 7373, CNRS, Aix-Marseille univ., Marseille, 13453, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A note on flatness of non separable tangent cone at a barycenter},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {489--495},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {4},
     year = {2020},
     doi = {10.5802/crmath.66},
     language = {en},
}

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Thibaut Le Gouic. A note on flatness of non separable tangent cone at a barycenter. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 489-495. doi : 10.5802/crmath.66. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.66/

Bibliographie
Cité par

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[2] Stephanie Alexander; Vitali Kapovitch; Anton Petrunin Alexandrov geometry (2019) (http://arxiv.org/abs/1903.08539) | Zbl

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[6] Thibaut Le Gouic; Quentin Paris; Philippe Rigollet; Austin J. Stromme Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space (2019) (https://arxiv.org/abs/1908.00828)

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[8] Karl-Theodor Sturm Metric spaces of lower bounded curvature, Expo. Math., Volume 17 (1999) no. 1, pp. 35-47 | MR | Zbl

[9] Takumi Yokota A rigidity theorem in Alexandrov spaces with lower curvature bound, Math. Ann., Volume 353 (2012) no. 2, pp. 305-331 | DOI | MR | Zbl

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