Electronic Communications in Probability

Gaussian measures of dilations of convex rotationally symmetric sets in $\mathbb{C}^n$

Tomasz Tkocz

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Abstract

We consider the complex case of the S-inequality. It concerns the behaviour of Gaussian measures of dilations of convex and rotationally symmetric sets in $\mathbb{C}^n$. We pose and discuss a conjecture that among all such sets measures of cylinders decrease the fastest under dilations. Our main result in this paper is that this conjecture holds under the additional assumption that the Gaussian measure of the sets considered is not greater than some constant $c > 0.64$.

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 5, 38-49.

Dates
Accepted: 12 January 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261961

Digital Object Identifier
doi:10.1214/ECP.v16-1599

Mathematical Reviews number (MathSciNet)
MR2763527

Zentralblatt MATH identifier
1225.60059

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G15: Gaussian processes

Keywords
Gaussian measure convex bodies isoperimetric inequalities

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Tkocz, Tomasz. Gaussian measures of dilations of convex rotationally symmetric sets in $\mathbb{C}^n$. Electron. Commun. Probab. 16 (2011), paper no. 5, 38--49. doi:10.1214/ECP.v16-1599. https://projecteuclid.org/euclid.ecp/1465261961


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References

  • A. Ehrhard. Symetrisation dans l'espace de Gauss. Math. Scand. 53 (1983), no. 2, 281–301.
  • R. J. Gardner. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405.
  • K. Ito and H. P. McKean, Jr.. Diffusion processes and their sample paths. Die Grundlehren der Mathematischen Wissenschaften, Band 125 Academic Press, Publishers, New York, 1965.
  • S. Kwapien and J. Sawa. On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets. Studia Math. 105 (1993), no. 2, 173–187.
  • R. Latala and K. Oleszkiewicz. Gaussian measures of dilatations of convex symmetric sets. Ann. Probab. 27 (1999), no. 4, 1922–1938.