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Stability of Talagrand's inequality under concentration topology

Authors: Ryunosuke Ozawa and Norihiko Suzuki
Journal: Proc. Amer. Math. Soc. 145 (2017), 4493-4501
MSC (2010): Primary 53C23; Secondary 60E15
Published electronically: April 27, 2017
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Abstract: In this paper, we study the compatibility between Talagrand's inequality and the concentration topology; i.e., if a sequence of mm-spaces satisfying Talagrand's inequality converges with respect to the observable distance, then the limit space satisfies Talagrand's inequality.

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  • [1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer, Sur les inégalités de Sobolev logarithmiques, (French, with French summary), with a preface by Dominique Bakry and Michel Ledoux, Panoramas et Synthèses [Panoramas and Syntheses], vol. 10, Société Mathématique de France, Paris, 2000. MR 1845806
  • [2] V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655
  • [3] Kei Funano and Takashi Shioya, Concentration, Ricci curvature, and eigenvalues of Laplacian, Geom. Funct. Anal. 23 (2013), no. 3, 888-936. MR 3061776,
  • [4] Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 1699320
  • [5] John Lott and Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903-991. MR 2480619,
  • [6] Takashi Shioya, Metric measure geometry, Gromov's theory of convergence and concentration of metrics and measures, IRMA Lectures in Mathematics and Theoretical Physics, vol. 25, EMS Publishing House, Zürich, 2016. MR 3445278
  • [7] Karl-Theodor Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65-131. MR 2237206,
  • [8] M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal. 6 (1996), no. 3, 587-600. MR 1392331,
  • [9] Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483
  • [10] Cédric Villani, Optimal transport, Old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. MR 2459454

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Additional Information

Ryunosuke Ozawa
Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Address at time of publication: Department of Mathematics, Osaka University, Osaka 560-0043, Japan

Norihiko Suzuki
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Keywords: Metric measure space, observable distance, Talagrand's inequality
Received by editor(s): July 18, 2016
Received by editor(s) in revised form: November 1, 2016
Published electronically: April 27, 2017
Additional Notes: The first author was supported by JSPS KAKENHI Grant No. 24224002 and postdoctoral program at Max Planck Institute for Mathematics.
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2017 American Mathematical Society

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