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Polar Isoperimetry. I: The Case of the Plane

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High Dimensional Probability VIII

Part of the book series: Progress in Probability ((PRPR,volume 74))

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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.

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References

  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+ 552

    Google Scholar 

  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+ 331

    Google Scholar 

  3. A. Dembo, T.M. Cover, J.A. Thomas, Information-theoretic inequalities. IEEE Trans. Inform. Theory 37(6), 1501–1518 (1991)

    Article  MathSciNet  Google Scholar 

  4. A. Diaz, N. Harman, S. Howe, D. Thompson, Isoperimetric problems in sectors with density. Adv. Geom. 12(4), 589–619 (2012)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  6. T. van Erven, P. Harremoës, Rényi divergence and Kullback-Leibler divergence. IEEE Trans. Inform. Theory 60(7), 3797–3820 (2014)

    Article  MathSciNet  Google Scholar 

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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.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Sergey G. Bobkov

  2. Université Paris Descartes, MAP5, UMR 8145, Paris Cedex, France

    Nathael Gozlan

  3. Université Paris Nanterre, MODAL’X, EA 3454, Nanterre, France

    Cyril Roberto

  4. LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447, Marne-la-Vallée, France

    Paul-Marie Samson

Authors
  1. Sergey G. Bobkov
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  2. Nathael Gozlan
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  3. Cyril Roberto
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  4. Paul-Marie Samson
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Corresponding author

Correspondence to Sergey G. Bobkov .

Editor information

Editors and Affiliations

  1. MAP 5, Université Paris Descartes, Paris, France

    Nathael Gozlan

  2. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Rafał Latała

  3. Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France

    Karim Lounici

  4. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Mokshay Madiman

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Bobkov, S.G., Gozlan, N., Roberto, C., Samson, PM. (2019). Polar Isoperimetry. I: The Case of the Plane. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_3

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