The inequality \[ \| u \|_{W_2^2 (\mathcal{R})} \leqq \left( {1 + \frac{1}{\lambda } + \frac{1}{{\lambda ^2 }}} \right)^{{1 / 2}} \| {\Delta u} \|_{L_2 (\mathcal{R})} ,\] for $u \in W_{2,0}^2 (\mathcal{R})$, is presented. The (n-dimensional) region $\mathcal{R}$ has a piecewise smooth boundary with nonnegative mean curvature, and $\lambda $ is the fundamental frequency of $\mathcal{R}$. If $\mathcal{R}$ is a convex polyhedron, the inequality is sharp, so that the fundamental frequency of $\mathcal{R}$ is characterized.