  Simultaneous measurements of position and momentum are considered in $n$
dimensions. We find, that for a particle whose position is strictly localized
in a compact domain $D\subset \mathbb{R}^n$ (spatial uncertainty) with
non-empty boundary, the standard deviation of its momentum is sharply bounded
by $\sigma_p \geq \lambda_1^{1/2}\hbar$, while $\lambda_1$ is the first
Dirichlet eigenvalue of the Laplacian on $D$.
