locating the first nodal set in higher dimensions

Abstract: Let $\Omega$ be a bounded convex domain in $\R^n$. Let $u$ be the
Neumann eigenfunction for $\Omega$ associated with the smallest nonzero
eigenvalue $\lambda$, that is, $$ \Delta u = -\lambda u \hbox{ on } \Omega,
u_{\nu} = 0 \hbox{ on } \partial \Omega.$$ In a joint work with David
Jerison and Inwon Kim, we give an estimate for the location of the nodal
set $\Lambda = \{ u=0\}$ in terms of the eccentricity
 of the domain.  Corresponding results for planar domains ($n=2$)
have been previously proved by David Jerison.