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Spectral Gap for log-Concave Probability Measures on the Real Line

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1857)

Abstract

We propose a simple parameter to describe the exact order of the Poincaré constant (or the inverse of the spectral gap) for a log-concave probability measure on the real line. This parameter is the square of the mean value of the distance to the median. Bobkov recently derived a similar result in terms of the variance of the measure. His approach was based on the study of the Cheeger constant. Our viewpoint is quite different and makes use of the Muckenhoupt functional and of a variational computation in the set of convex functions.

Keywords

  • Convex Function
  • Real Line
  • Isoperimetric Inequality
  • Good Constant
  • Logarithmic Sobolev Inequality

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Pierre Fougères: Address during the work: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse, France

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Correspondence to Pierre Fougères .

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© 2005 Springer-Verlag Berlin/Heidelberg

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Fougères, P. (2005). Spectral Gap for log-Concave Probability Measures on the Real Line. In: Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics, vol 1857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31449-3_7

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