The uncertainty principle in terms of isoperimetric inequalities

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 $D$.