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Discrete isoperimetric and Poincaré-type inequalities
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  • Published: May 1999

Discrete isoperimetric and Poincaré-type inequalities

  • S. G. Bobkov1 &
  • F. Götze2 

Probability Theory and Related Fields volume 114, pages 245–277 (1999)Cite this article

  • 327 Accesses

  • 38 Citations

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Abstract

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μn on the discrete cube {0, 1}n and on the lattice Z n. In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z n. The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes.

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Authors and Affiliations

  1. Department of Mathematics, Syktyvkar University, 167001 Syktyvkar, Russia, , , , , , RU

    S. G. Bobkov

  2. Department of Mathematics, Bielefeld University, P.O. Box 100131, D-33501, Bielefeld, Germany. e-mail: goetze@mathematik.uni-bielefeld.de, , , , , , DE

    F. Götze

Authors
  1. S. G. Bobkov
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  2. F. Götze
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Received: 30 April 1997 / Revised version: 5 June 1998

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Bobkov, S., Götze, F. Discrete isoperimetric and Poincaré-type inequalities. Probab Theory Relat Fields 114, 245–277 (1999). https://doi.org/10.1007/s004400050225

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  • Issue Date: May 1999

  • DOI: https://doi.org/10.1007/s004400050225

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  • Mathematics Subject Classification (1991): Primary 60E15; Secondary 26D15
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