Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain. Denote by $\{\lambda_{k}\}_{k=1}^{\infty}$ (resp. $\{\mu_{k}\}_{k=1}^{\infty})$ the eigenvalues of the Laplace operator in $\Omega$ with Dirichlet (resp. Neumann) boundary conditions. Introduce notations
$\Phi(d,k,\Omega)=\#\{j:\mu_{j}(\Omega)\le\lambda_{k}(\Omega)\}$, $\Psi(d,k,\Omega)=\Phi(d,k,\Omega)-k$ ,
so the inequality
$\mu_{k+\Psi(d,k,\Omega)}\le\lambda_{k}$
holds true. In 1986, Levine and Weinberger proved the estimate $\Psi(d,k,\Omega)\ge d$ for all convex domains. We show that for $d\gg1$ this result can be improved:
$\Psi(d,k,\Omega)\ge C(\frac{e}{2})^{d}$
also for all convex domains. The similar estimate holds for $k=1$:
$\Psi(d,1,\Omega)\ge C(\frac{e}{2})^{d}$
for arbitrary domains.