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

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 :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.66
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
@article{CRMATH_2020__358_4_489_0,
     author = {Thibaut Le Gouic},
     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},
}
TY  - JOUR
AU  - Thibaut Le Gouic
TI  - A note on flatness of non separable tangent cone at a barycenter
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 489
EP  - 495
VL  - 358
IS  - 4
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.66
LA  - en
ID  - CRMATH_2020__358_4_489_0
ER  - 
%0 Journal Article
%A Thibaut Le Gouic
%T A note on flatness of non separable tangent cone at a barycenter
%J Comptes Rendus. Mathématique
%D 2020
%P 489-495
%V 358
%N 4
%I Académie des sciences, Paris
%R 10.5802/crmath.66
%G en
%F CRMATH_2020__358_4_489_0
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/

[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

Cité par Sources :

Commentaires - Politique


Ces articles pourraient vous intéresser

Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds

Giuseppe Savaré

C. R. Math (2007)


Vers un théorème de la limite centrale dans l'espace de Wasserstein ?

Martial Agueh; Guillaume Carlier

C. R. Math (2017)


An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities

Sinho Chewi; Aram-Alexandre Pooladian

C. R. Math (2023)