We study the first positive Neumann eigenvalue $\mu_1$ of the
Laplace operator on a planar domain $\Omega$. We are particularly interested in
how the size of $\mu_1$ depends on the size and geometry of $\Omega$.
A notion of the intrinsic diameter of $\Omega$ is proposed and
various examples are provided to illustrate the effect of the
intrinsic diameter and its interplay with the geometry of the