Exact lower and upper bounds for Gaussian measures

Exact upper and lower bounds on the ratio $\operatorname{\mathbf{E}}w(\mathbf{X}-\mathbf{v})/\operatorname{\mathbf{E}}w(\mathbf{X})$ for a centered Gaussian random vector $\mathbf{X}$ in $\mathbf{R}^n$ are obtained, as well as bounds on the rate of change of $\operatorname{\mathbf{E}}w(\mathbf{X}-t\mathbf{v})$ in $t$, where $w\colon\mathbf{R}^n\to[0,\infty)$ is any even unimodal function and $\mathbf{v}$ is any vector in $\mathbf{R}^n$. As a corollary of such results, exact upper and lower bounds on the power function of statistical tests for the mean of a multivariate normal distribution are given.