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Polar Isoperimetry. I: The Case of the Plane

  • Sergey G. BobkovEmail author
  • Nathael Gozlan
  • Cyril Roberto
  • Paul-Marie Samson
Conference paper
Part of the Progress in Probability book series (PRPR, volume 74)

Abstract

This is the first part of the notes with preliminary remarks on the plane isoperimetric inequality and its applications to the Poincaré and Sobolev-type inequalities in dimension one. Links with informational quantities of Rényi and Fisher are briefly discussed.

Keywords

Isoperimetry Sobolev-type inequalities Rényi divergence power Relative Fisher information 

Notes

Acknowledgements

Research was partially supported by the NSF grant DMS-1855575 and by the Bzout Labex, funded by ANR, reference ANR-10-LABX-58, the Labex MME-DII funded by ANR, reference ANR-11-LBX-0023-01, and the ANR Large Stochastic Dynamic, funded by ANR, reference ANR-15-CE40-0020-03-LSD.

References

  1. 1.
    D. Bakry, I. Gentil, M. Ledoux, Analysis and geometry of Markov diffusion operators, in Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 348 (Springer, Cham, 2014), p. xx+ 552Google Scholar
  2. 2.
    Yu.D. Burago, V.A. Zalgaller, Geometric inequalities, in Translated from the Russian by A. B. Sosinskii. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer Series in Soviet Mathematics, vol. 285 (Springer, Berlin, 1988), p. xiv+ 331Google Scholar
  3. 3.
    A. Dembo, T.M. Cover, J.A. Thomas, Information-theoretic inequalities. IEEE Trans. Inform. Theory 37(6), 1501–1518 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Diaz, N. Harman, S. Howe, D. Thompson, Isoperimetric problems in sectors with density. Adv. Geom. 12(4), 589–619 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    E. Lutwak, D. Yang, G. Zhang, Cramer-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information. IEEE Trans. Inform. Theory 51(2), 473–478 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    T. van Erven, P. Harremoës, Rényi divergence and Kullback-Leibler divergence. IEEE Trans. Inform. Theory 60(7), 3797–3820 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sergey G. Bobkov
    • 1
    Email author
  • Nathael Gozlan
    • 2
  • Cyril Roberto
    • 3
  • Paul-Marie Samson
    • 4
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Université Paris Descartes, MAP5, UMR 8145Paris CedexFrance
  3. 3.Université Paris Nanterre, MODAL’X, EA 3454NanterreFrance
  4. 4.LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447Marne-la-ValléeFrance

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