  We study the Dirichlet eigenvalues of the Laplacian on a convex domain in
$\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper
bounds for the Riesz means of order $\sigma\geq 3/2$ established in an article
by Geisinger, Laptev and Weidl. This is achieved by refining estimates for a
negative second term in the Berezin inequality. The obtained remainder term
reflects the correct order of growth in the semi-classical limit and depends
only on the measure of the boundary of the domain. We emphasize that such an
improvement is for general $\Omega\subset\mathbb{R}^n$ not possible and was
previously known to hold only for planar convex domains satisfying certain
geometric conditions.
  As a corollary we obtain lower bounds for the individual eigenvalues
$\lambda_k$, which for a certain range of $k$ improves the Li--Yau inequality
for convex domains. However, for convex domains one can use different methods
to obtain even stronger such lower bounds.
