Étant donné une mesure de probabilité sur un espace d’Alexandrov avec courbure minorée, nous prouvons que le support de la mesure poussée de sur le cône tangent à son barycentre (exponentiel) est un sous-ensemble d’un espace de Hilbert, sans condition de séparabilité du cône tangent.
Given a probability measure on an Alexandrov space with curvature bounded below, we prove that the support of the pushforward of on the tangent cone at its (exponential) barycenter is a subset of a Hilbert space, without separability of the tangent cone.
@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}, }
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] On the rate of convergence of empirical barycentres in metric spaces: curvature, convexity and extendible geodesics (2018) (https://arxiv.org/abs/1806.02740v1)
[2] Alexandrov geometry (2019) (http://arxiv.org/abs/1903.08539) | Zbl
[3] A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, 2001 | MR | Zbl
[4] On tangent cones of Alexandrov spaces with curvature bounded below, Manuscr. Math., Volume 103 (2000) no. 2, pp. 169-182 | DOI | MR | Zbl
[5] Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal., Volume 7 (1997) no. 3, pp. 535-560 | DOI | MR | Zbl
[6] Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space (2019) (https://arxiv.org/abs/1908.00828)
[7] Barycenters in Alexandrov spaces of curvature bounded below, Adv. Geom., Volume 12 (2012) no. 4, pp. 571-587 | MR | Zbl
[8] Metric spaces of lower bounded curvature, Expo. Math., Volume 17 (1999) no. 1, pp. 35-47 | MR | Zbl
[9] 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
Vers un théorème de la limite centrale dans l'espace de Wasserstein ?
Martial Agueh; Guillaume Carlier
C. R. Math (2017)
Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds
Giuseppe Savaré
C. R. Math (2007)
An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities
Sinho Chewi; Aram-Alexandre Pooladian
C. R. Math (2023)