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

The uncertainty principle in terms of isoperimetric inequalities