  We establish a method for giving lower bounds for the fundamental tone of
elliptic operators in divergence form in terms of the divergence of vector
fields. We then apply this method to the $L_{r}$ operator associated to
immersed hypersurfaces with locally bounded $(r+1)$-th mean curvature $H_{r+1}$
of the space forms $\mathbb{N}^{n+1}(c)$ of curvature $c$. As a corollary we
give lower bounds for the extrinsic radius of closed hypersurfaces of
$\mathbb{N}^{n+1}(c)$ with $H_{r+1}>0$ in terms of the $r$-th and $(r+1)$-th
mean curvatures. Finally we observe that bounds for the Laplace eigenvalues
essentially bound the eigenvalues of a self-adjoint elliptic differential
operator in divergence form. This allows us to show that Cheeger's constant
gives a lower bounds for the first nonzero $L_{r}$-eigenvalue of a closed
hypersurface of $\mathbb{N}^{n+1}(c)$.
