  We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace
eigenvalues in domains that admit a representation in Fermi coordinates. More
precisely, if $\gamma$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric
with respect to the $y$-axis, let $D\subset \mathbb{R}^2$ denote the domain of
points that lie on one side of $\gamma$ and within a prescribed distance
$\delta(s)$ from $\gamma(s)$ (here $s$ denotes the arc length parameter for
$\gamma$). Write $\mu_1^{odd}(D)$ for the lowest nonzero eigenvalue of the
Neumann $p$-Laplacian with an eigenfunction that is odd with respect to the
$y$-axis. For all $p>1$, we provide a lower bound on $\mu_1^{odd}(D)$ when the
distance function $\delta$ and the signed curvature $k$ of $\gamma$ satisfy
certain geometric constraints. In the linear case ($p=2$), we establish
sufficient conditions to guarantee $\mu_1^{odd}(D)=\mu_1(D)$. We finally study
the asymptotics of $\mu_1(D)$ as the distance function tends to zero. We show
that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of
a weighted one-dimensional Neumann $p$-Laplace problem.
