  For an $n$-dimensional polytope $\Omega$ in $\mathbb{R}^{n}$, we study lower
bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In
the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau
(1983) obtained the first term with the order $k^{\frac2n}$, which is optimal.
The next landmark goal is to give the second term with the order $k^{\frac1n}$
in the asymptotic formula. For this purpose, Kova\v{r}\'{\i}k, Vugalter and
Weidl (2009) have made an important breakthrough in the case of dimension 2. It
is our purpose to study the $n$-dimensional case for arbitrary dimension $n$.
We obtain the second term in the asymptotic sense.
