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.
References
Borell Ch. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30: 207–216
Borell Ch. (2003). The Ehrhard inequality. C. R. Acad. Sci. Paris Ser. I 337: 663–666
Borell, Ch.: Minkowski sums and Brownian exit times. Ann. Fac. Sci. Toulouse (in press)
Dudley R.M. (1989). Real analysis and probability. Wadsworth/Brooks-Cole, Pacific Grove
Ehrhard A. (1983). Symétrisation dans l’espace de Gauss. Math. Scand. 53: 281–301
Latała R.A. (1996). A note on Ehrhard’s inequality. Stud Math 118: 169–174
Latała R.A. (2002). On some inequalities for Gaussian measures. Proc. ICM 2: 813–822
Sudakov, V.N., Tsirelson, B.S.: Extremal properties of half-spaces for spherically invariant measures. Zap. Nauch. Sem. L.O.M.I. 41, 14–24 (1974) [translated in J. Soviet Math. 9, 9–18 (1978)]
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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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DOI: https://doi.org/10.1007/s00440-007-0062-5
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