Bounding Eigenvalues with Packing Density

We prove a lower bound on the eigenvalues $\lambda_k$, $k\in\mathbb{N}$, of the Dirichlet Laplacian of a bounded domain $\Omega\subset\mathbb{R}^n$ of volume $V$: $$\lambda_k \geq C_n\bigg( \delta\frac{k}{V}\bigg)^{2/n}$$ where $\delta$ is a constant that measures how efficiently $\Omega$ can be packed into $\mathbb{R}^n$ and $C_n$ is the constant found in Weyl's law. This generalizes a result of Urakawa in 1984. If $\delta^{2/n} > n/(n+2)$, this bound is stronger than the eigenvalue bound proven by Li and Yau in 1983. For example, in the case of convex planar domains, we have for all $k\in\mathbb{N}$, $$\lambda_k \geq \frac{2\sqrt{3}\pi k}{V}.$$