We will discuss lower bounds for the first non-trivial Neumann eigenvalue $\mu_1(\Omega)$ of the $p$-Laplace operator ($p \gt 1$) in a Lipschitz, bounded domain $\Omega$ in $\mathbb{R}^n$. In 1960 Payne and Weinberger proved that, when $\Omega$ is convex and $p = 2$, then \begin{equation}\label{eq:1}\mu_1(\Omega) \geq \frac{\pi^2}{d(\Omega)^2},\end{equation} where $d(\Omega)$ is the diameter of $\Omega$. The above estimate is asymptotically sharp, since $\mu_1(\Omega)d(\Omega)^2$ tends to $\pi^2$ for a parallelepiped all but one of whose dimensions shrink to $0$. On the other hand, it does not hold true in general for non-convex sets. In this talk we will focus on the non-convex setting. We will consider an arbitrary Lipschitz, bounded domain $\Omega$ in $\mathbb{R}^n$ and we will show a sharp lower bound for $\mu_1(\Omega)$ which, differently from \eqref{eq:1}, involves the best isoperimetric constant relative to $\Omega$ and is sharp, at least when $p = n = 2$, as the isoperimetric constant relative to $\Omega$ goes to $0$. Moreover, in a suitable class of convex planar domains, our estimate will turn out to be better than \eqref{eq:1}.

Furthermore, we will see that, when $p = n = 2$ and $\Omega$ consists of the points on one side of a smooth curve $\gamma$, within a suitable distance $\delta$ from it, then $\mu_1(\Omega)$ can be sharply estimated from below in terms of the length of $\gamma$, the $L^\infty$ norm of its curvature and $\delta$.