  We study the lower bounds for the principal frequency of the $p$-Laplacian on
$N$-dimensional Euclidean domains. For $p>N$, we obtain a lower bound for the
first eigenvalue of the $p$-Laplacian in terms of its inradius, without any
assumptions on the topology of the domain. Moreover, we show that a similar
lower bound can be obtained if $p > N-1$ assuming the boundary is connected.
This result can be viewed as a generalization of the classical bounds for the
first eigenvalue of the Laplace operator on simply connected planar domains.
