We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $R^d$, $d geq 2$. In particular, we derive upper bounds on Riesz means of order $sigma geq 3/2$, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit. Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.
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