  Let $\Omega$ be a smooth, convex, unbounded domain of $\R^N$. Denote by
$\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator
in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since
equality sign is achieved when $\Omega$ is a $N$-dimensional strip. Our
estimate can be equivalently viewed as an optimal Poincar\'e-Wirtinger
inequality for functions belonging to the weighted Sobolev space
$H^1(\Omega,d\gamma_N)$, where $\gamma_N$ is the $N$-dimensional Gaussian
measure.
