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},
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     pages = {489--495},
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     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

[1] Adil Ahidar-Coutrix; Thibaut Le Gouic; Quentin Paris On the rate of convergence of empirical barycentres in metric spaces: curvature, convexity and extendible geodesics (2018) (https://arxiv.org/abs/1806.02740v1)

[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)

[7] Shin-Ichi Ohta Barycenters in Alexandrov spaces of curvature bounded below, Adv. Geom., Volume 12 (2012) no. 4, pp. 571-587 | MR | Zbl

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Yueqi Cao; Anthea Monod A geometric condition for uniqueness of Fréchet means of persistence diagrams, Computational Geometry, Volume 128 (2025), p. 15 (Id/No 102162) | DOI:10.1016/j.comgeo.2024.102162 | Zbl:8008642
Thibaut Le Gouic; Quentin Paris; Philippe Rigollet; Austin J. Stromme Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space, Journal of the European Mathematical Society (JEMS), Volume 25 (2023) no. 6, pp. 2229-2250 | DOI:10.4171/jems/1234 | Zbl:7714611
A. Ahidar-Coutrix; T. Le Gouic; Q. Paris Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics, Probability Theory and Related Fields, Volume 177 (2020) no. 1-2, pp. 323-368 | DOI:10.1007/s00440-019-00950-0 | Zbl:1442.51004

Cité par 3 documents. Sources : zbMATH

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