## The Annals of Probability

### Gaussian Measures of Dilatations of Convex Symmetric Sets

#### Abstract

We prove that the inequality $\Psi^-1(\mu(tA))\geq t\Psi^-1(\mu(A))$ holds for any centered Gaussian measure $\mu$ on a separable Banach space $F$, any convex, closed, symmetric set $A\subset{F}$ and $t\geq1$, where $\Psi(x)=\gamma_1(-x,x)=(2\pi)^-1/2\int_{-x}^x\exp(-y^2)2)dy$. As an application, the best constants in comparison of moments of Gaussian vectors are calculated.

#### Article information

Source
Ann. Probab., Volume 27, Number 4 (1999), 1922-1938.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022874821

Digital Object Identifier
doi:10.1214/aop/1022874821

Mathematical Reviews number (MathSciNet)
MR20001h:60026

Zentralblatt MATH identifier
0966.60037

#### Citation

Latała, Rafał; Oleszkiewicz, Krzysztof. Gaussian Measures of Dilatations of Convex Symmetric Sets. Ann. Probab. 27 (1999), no. 4, 1922--1938. doi:10.1214/aop/1022874821. https://projecteuclid.org/euclid.aop/1022874821

#### References

• [1] Ehrhard, A. (1983). Symetrisation dans l'espace de Gauss. Math. Scand. 53 281-301.
• [2] Hitczenko, P., Kwapie ´n, S., Li, W., Schechtman, G., Schlumprecht, T. and Zinn, J. (1998). Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables. Electron. J. Probab. 3.
• [3] It o K. and McKean H. P. (1965). Diffusion Processes and Their Sample Paths. Springer, New York.
• [4] Kwapie ´n, S. and Sawa, J. (1993). On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets. Studia Math. 105 173-187.
• [5] Sudakov, V. N. and Zalgaller, V. A. (1974). Some problems on centrally symmetric convex bodies. In Problems in Global Geometry. Zap. Nau cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 45 75-82, 119 (in Russian).
• [6] Szarek, S. (1991). Condition numbers of random matrices. J. Complexity 7 131-149.