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

Thibaut Le Gouic

doi:10.5802/crmath.66

Geometry, Probability Theory

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

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.

Abstract
Résumé

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.

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.

Received: 2019-09-16
Revised: 2019-12-31
Accepted: 2020-04-29
Published online: 2020-07-28

DOI: 10.5802/crmath.66

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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},
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     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/

References
Cited by

[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

[3] Dmitri Burago; Yuri Burago; Sergei Ivanov A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, 2001 | MR | Zbl

[4] Stephanie Halbeisen On tangent cones of Alexandrov spaces with curvature bounded below, Manuscr. Math., Volume 103 (2000) no. 2, pp. 169-182 | DOI | MR | Zbl

[5] Urs Lang; Viktor Schroeder Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal., Volume 7 (1997) no. 3, pp. 535-560 | DOI | MR | Zbl

[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

[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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