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.