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Inequalities of the Brunn–Minkowski type for Gaussian measures
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  • Published: 22 February 2007

Inequalities of the Brunn–Minkowski type for Gaussian measures

  • Christer Borell1,2 

Probability Theory and Related Fields volume 140, pages 195–205 (2008)Cite this article

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  • 11 Citations

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Abstract

Let \(m\geq 2\) be an integer, let γ be the standard Gaussian measure on \(\mathbf{R}^{n}\), and let \(\Phi (t)=\int_{-\infty }^{t}\exp (-s^{2}/2)ds/ \sqrt{2\pi }{\small ,} -\infty \leq t\leq \infty\). Given \(\alpha _{1},\ldots,\alpha _{m}\in \left] 0,\infty \right[ \) this paper gives a necessary and sufficient condition such that the inequality \(\Phi ^{-1}(\gamma (\alpha _{1}A_{1}+\cdots+\alpha _{m}A_{m}))\geq \alpha _{1}\Phi ^{-1}(\gamma (A_{1}))+\cdots+ \alpha _{m}\Phi ^{-1}(\gamma (A_{m}))\) is true for all Borel sets A 1,...,A m in \(\mathbf{R}^{n}\) of strictly positive γ-measure or all convex Borel sets A 1,...,A m in \(\mathbf{R}^{n}\) of strictly positive γ-measure, respectively. In particular, the paper exhibits inequalities of the Brunn–Minkowski type for γ which are true for all convex sets but not for all measurable sets.

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References

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Authors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology, 412 96, Göteborg, Sweden

    Christer Borell

  2. Department of Mathematical Sciences, Göteborg University, 412 96, Göteborg, Sweden

    Christer Borell

Authors
  1. Christer Borell
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Correspondence to Christer Borell.

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Cite this article

Borell, C. Inequalities of the Brunn–Minkowski type for Gaussian measures. Probab. Theory Relat. Fields 140, 195–205 (2008). https://doi.org/10.1007/s00440-007-0062-5

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  • Received: 15 August 2006

  • Revised: 11 January 2007

  • Published: 22 February 2007

  • Issue Date: January 2008

  • DOI: https://doi.org/10.1007/s00440-007-0062-5

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Keywords

  • Gauss measure
  • Linear combination of sets
  • Inequality of the Brunn–Minkowski type
  • Measurable set
  • Convex set

Mathematics Subject Classification (2000)

  • 60B05
  • 60D05
  • 60E15
  • 60G15
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