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Probability Theory and Related Fields

, Volume 140, Issue 1–2, pp 195–205 | Cite as

Inequalities of the Brunn–Minkowski type for Gaussian measures

  • Christer BorellEmail author
Article

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.

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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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of TechnologyGöteborgSweden
  2. 2.Department of Mathematical SciencesGöteborg UniversityGöteborgSweden

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