  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.
