no. 4
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]
Le Gouic, Thibaut1 
1 Massachusetts Institute of Technology, Department of Mathematics and Centrale Marseille, I2M, UMR 7373, CNRS, Aix-Marseille univ., Marseille, 13453, France
Comptes Rendus. Mathématique, Tome 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
Le Gouic, Thibaut 1

1 Massachusetts Institute of Technology, Department of Mathematics and Centrale Marseille, I2M, UMR 7373, CNRS, Aix-Marseille univ., Marseille, 13453, France 
@article{CRMATH_2020__358_4_489_0,
     author = {Le Gouic, Thibaut},
     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},
     url = {http://www.numdam.org/articles/10.5802/crmath.66/}
}
TY  - JOUR
AU  - Le Gouic, Thibaut
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
UR  - http://www.numdam.org/articles/10.5802/crmath.66/
DO  - 10.5802/crmath.66
LA  - en
ID  - CRMATH_2020__358_4_489_0
ER  - 
%0 Journal Article
%A Le Gouic, Thibaut
%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
%U http://www.numdam.org/articles/10.5802/crmath.66/
%R 10.5802/crmath.66
%G en
%F CRMATH_2020__358_4_489_0
Le Gouic, Thibaut. A note on flatness of non separable tangent cone at a barycenter. Comptes Rendus. Mathématique, Tome 358 (2020) no. 4, pp. 489-495. doi : 10.5802/crmath.66. http://www.numdam.org/articles/10.5802/crmath.66/

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

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

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

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

[5] Lang, Urs; Schroeder, Viktor Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal., Volume 7 (1997) no. 3, pp. 535-560 (Accessed 2019-05-15) | DOI | MR | Zbl

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

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

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

[9] Yokota, Takumi 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 :