Locating the first nodal set in higher dimensions

We extend the two-dimensional results of Jerison (2000) on the location of the nodal set of the first Neumann eigenfunction of a convex domain to higher dimensions. If a convex domain $\Omega$ in $\mathbb{R}^n$ is contained in a long and thin cylinder $[0,N] \times B_{\epsilon}(0)$ with nonempty intersections with $\{x_1= 0\}$ and $\{x_1=N\}$, then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation, by a bound proportional to $\epsilon$, whose coefficients are expressed in terms of the volume of the cross sections of the domain. Also, the first nodal set is located within a distance comparable to $\epsilon$ near the zero of the corresponding ordinary differential equation.